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Topics and sample problems for Math 017 Final Exam (long version) Below there are 59 categories: 9 “arithmetic” categories 13 “easy algebra” categories 24 “basic algebra” categories 9 “difficult algebra” categories 4 “very difficult algebra” categories Each version of the exam will be created as follows: First, 45 categories will be selected: All 9 arithmetic categories will be included. From the algebraic categories, 8 categories will be selected at random from 13 “easy” categories, all 24“basic algebra” categories will be represented, 3 out of the 9 “difficult” categories, and 1 from the“very difficult” category. After this process, we will have 45 categories. One question out of each such category will be selected at random. Thus, the test will consist of 45 questions. ARITHMETIC PART: 9 QUESTIONS Select randomly one question from each category. 1A Addition, subtraction, multiplication, or division of integers (using both: and fraction bar as a division sign) - up to 3 integers. Evaluate, if possible. Otherwise, write “undefined”: 3 5 Compute, if possible. Otherwise, write “undefined”: 7 1 Evaluate, if possible. Otherwise, write “undefined”: 3 10 Compute, if possible. Otherwise, write “undefined”: 3 (2) (11) Evaluate, if possible. Otherwise, write “undefined”: (1)( 4) Compute, if possible. Otherwise, write “undefined”: 2 (1) (1) Evaluate, if possible. Otherwise, write “undefined”: 10 2 1 Compute, if possible. Otherwise, write “undefined”: 10 2 Evaluate, if possible. Otherwise, write “undefined”: 48 8 Compute, if possible. Otherwise, write “undefined”: 3 (3) Evaluate, if possible. Otherwise, write “undefined”: 2(3)(7) Compute, if possible. Otherwise, write “undefined”: 4 (100) 100 Evaluate, if possible. Otherwise, write “undefined”: 2 5 9 Compute, if possible. Otherwise, write “undefined”: 6(1)( 2) Evaluate, if possible. Otherwise, write “undefined”: 4(10)( 100) Compute, if possible. Otherwise, write “undefined”: 5(1)(8) Evaluate, if possible. Otherwise, write “undefined”: 6(8) Compute, if possible. Otherwise, write “undefined”: 5 4 Evaluate, if possible. Otherwise, write “undefined”: 2 (9) Compute, if possible. Otherwise, write “undefined”: 10 (3) Compute, if possible. Otherwise, write “undefined”: 9 0 Evaluate, if possible. Otherwise, write “undefined”: 0 10 Compute, if possible. Otherwise, write “undefined”: 10 0 Compute, if possible. Otherwise, write “undefined”: 0 5 2A Order of operations, integers only, two operations (only integers as an answer, no exponents). 2 Compute, if possible. Otherwise, write “undefined”: 7 2 3 Compute, if possible. Otherwise, write “undefined”: 4 (7 8) Evaluate, if possible. Otherwise, write “undefined”: 5 10 5 Evaluate, if possible. Otherwise, write “undefined”: 1 1(1) Compute, if possible. Otherwise, write “undefined”: 4 4(4) Evaluate, if possible. Otherwise, write “undefined”: (6 1)( 2 9) Compute, if possible. Otherwise, write “undefined”: 4 (12 12) Evaluate, if possible. Otherwise, write “undefined”: 2 (1 3) Compute, if possible. Otherwise, write “undefined”: 2 (4 5) (3) Evaluate, if possible. Otherwise, write “undefined”: [3 (7)]( 2) Compute, if possible. Otherwise, write “undefined”: 4 (1 3 2) Evaluate, if possible. Otherwise, write “undefined”: 4 (7) (2) Compute, if possible. Otherwise, write “undefined”: 8(2) 8 Evaluate, if possible. Otherwise, write “undefined”: 37 44 Compute, if possible. Otherwise, write “undefined”: 2 3 (2) Evaluate, if possible. Otherwise, write “undefined”: 12 3 2 Compute, if possible. Otherwise, write “undefined”: 8 (9 2) Evaluate, if possible. Otherwise, write “undefined”: 4 (2 5) Compute, if possible. Otherwise, write “undefined”: 12 (3 2) Evaluate, if possible. Otherwise, write “undefined”: 5 3 4 Evaluate, if possible. Otherwise, write “undefined”: 4 ( 2 2) 3 Compute, if possible. Otherwise, write “undefined”: 12 6 (6) Evaluate, if possible. Otherwise, write “undefined”: (5 5) 10 3A Addition and subtraction of fractions or of fractions and integers (positive and negative). Compute: 2 4 3 5 3 5 Compute: 5 6 Evaluate: 2 3 4 Evaluate: 1 4 3 Evaluate: Evaluate: Evaluate: Evaluate: Evaluate: Evaluate: Evaluate: Compute: Compute: Compute: 2 3 3 7 5 3 4 5 4 5 9 12 3 9 14 2 3 1 8 5 3 6 7 2 23 4 2 7 9 3 10 4 9 7 4 18 9 4 Compute: Compute: Compute: Compute: Compute: Compute: 5 1 7 3 5 14 7 2 3 3 11 2 3 9 7 2 6 3 4 5 9 12 4A Multiplication of fractions and of fractions and integers (positive and negative). Compute: 3 (10) 5 Evaluate: Compute: Evaluate: Compute: 345 1 10 5 2 345 2 12 15 8 (18) 9 11 4 1 x x 7 3 11 Evaluate: 5 4 4 2 6 35 Compute: 5 3 4 x 12 10 8 Evaluate: 24 1 4 7 6 32 5 Compute: 7 321 x 2 321 6 Evaluate: 2 12 15 Compute: 100 Evaluate: 24 19 3 8 Compute: 7 4 x 2 21 Evaluate: 10 22 11 5 Compute: 9 4 x 7 3 Evaluate: 5 6 4 10 Compute: 5 8 2 15 Evaluate: 8 10 x 7 9 Compute: 7 2 x 3 3 Evaluate: 5 9 4 2 23 27 5A Division (using both and fraction bar) of fractions and of fractions and integers (positive and negative). 6 Evaluate, if possible. Otherwise, write “undefined”: Compute, if possible. Otherwise, write “undefined”: 3 4 1 2 3 2 5 7 Evaluate, if possible. Otherwise, write “undefined”: 3 4 2 Compute, if possible. Otherwise, write “undefined”: 4 2 7 9 8 7 14 2 5 Compute, if possible. Otherwise, write “undefined”: 5 3 Evaluate, if possible. Otherwise, write “undefined”: 16 ( 8) 9 2 Compute, if possible. Otherwise, write “undefined”: 20 5 Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: 2 15 6 4 11 10 Compute, if possible. Otherwise, write “undefined”: 8 9 5 6 9 12 13 Evaluate, if possible. Otherwise, write “undefined”: Compute, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Compute, if possible. Otherwise, write “undefined”: 3 10 1 5 2 1 5 7 14 3 7 9 2 21 1 14 Evaluate, if possible. Otherwise, write “undefined”: Compute, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Compute, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Compute, if possible. Otherwise, write “undefined”: 2 1 5 5 2 3 3 15 10 4 6 9 3 13 6A “Zero” category(number/0; 0/number; addition of opposite numbers; multiplication by 0). “Ten” category: Multiplication and division of decimals and integers by powers of ten. Compute, if possible. Otherwise, write “undefined”: 234 1000 Evaluate, if possible. Otherwise, write “undefined”: 0.13 10 Compute, if possible. Otherwise, write “undefined”: 0.2 100 Evaluate, if possible. Otherwise, write “undefined”: 0 237 Compute, if possible. Otherwise, write “undefined”: 0 23 Compute, if possible. Otherwise, write “undefined”: 2667 2667 17 Evaluate, if possible. Otherwise, write “undefined”: 0 Evaluate, if possible. Otherwise, write “undefined”: 0.2 10 Evaluate, if possible. Otherwise, write “undefined”: 3.21 100 Evaluate, if possible. Otherwise, write “undefined”: 6 100 Evaluate, if possible. Otherwise, write “undefined”: 112 10 8 4.5 100 Evaluate, if possible. Otherwise, write “undefined”: 0.234 100 Compute, if possible. Otherwise, write “undefined”: 7.2 1000 0 Compute, if possible. Otherwise, write “undefined”: 100 Compute, if possible. Otherwise, write “undefined”: 0 0 Compute, if possible. Otherwise, write “undefined”: 101000 Compute, if possible. Otherwise, write “undefined”: 10 67.5 Compute, if possible. Otherwise, write “undefined”: 98 (98) Compute, if possible. Otherwise, write “undefined”: 100 0.3 Evaluate, if possible. Otherwise, write “undefined”: 7A Addition, subtraction, multiplication and division of decimals (and integers). Evaluate, if possible. Otherwise, write “undefined”: 0.1 2.1 Compute, if possible. Otherwise, write “undefined”: 0.1 0.999 Compute, if possible. Otherwise, write “undefined”: 3 0.02 Compute, if possible. Otherwise, write “undefined”: 0.1 0.01 Evaluate, if possible. Otherwise, write “undefined”: 0.33 (0.011) 2 0 .2 1 .2 Compute, if possible. Otherwise, write “undefined”: 0.018 49 Compute, if possible. Otherwise, write “undefined”: 0.21 Compute, if possible. Otherwise, write “undefined”: −0.5 (−0.0001) Evaluate, if possible. Otherwise, write “undefined”: 3.2 − (− 1.4) Evaluate, if possible. Otherwise, write “undefined”: − 0.8 − (−0.04) Compute, if possible. Otherwise, write “undefined”: − 0.9 − 0.2 − 0.4 Compute, if possible. Otherwise, write “undefined”: 7.22 + (− 0.002) Compute, if possible. Otherwise, write “undefined”: − (− 0.8) + (− 0.2) Compute, if possible. Otherwise, write “undefined”: 0.71 (5.1) 3.4 Compute, if possible. Otherwise, write “undefined”: −3.5 (−2) Compute, if possible. Otherwise, write “undefined”: − 79.3 0.001 Compute, if possible. Otherwise, write “undefined”: 4 (−0.005) Compute, if possible. Otherwise, write “undefined”: −0.2 (−0.3) ( −0.4) Evaluate, if possible. Otherwise, write “undefined”: 9 Compute, if possible. Otherwise, write “undefined”: 2 (−0.2) 0.03 8A Exponential notation (fractions, integers, decimals) Evaluate, if possible. Otherwise, write “undefined”: 16 Compute, if possible. Otherwise, write “undefined”: (0.1) 3 2 2 Evaluate, if possible. Otherwise, write “undefined”: 3 Evaluate, if possible. Otherwise, write “undefined”: (0.9) 2 Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: (0.4) 3 (1.1) 2 (0.01) 4 (0.06) 2 (0.1) 7 24 (1)14 (7) 2 (1) 25 Evaluate, if possible. Otherwise, write “undefined”: (3) 3 9 Evaluate, if possible. Otherwise, write “undefined”: 8 2 1 Evaluate, if possible. Otherwise, write “undefined”: 2 43 Evaluate, if possible. Otherwise, write “undefined”: 9 4 2 Evaluate, if possible. Otherwise, write “undefined”: 3 52 Evaluate, if possible. Otherwise, write “undefined”: 6 3 1 Evaluate, if possible. If not possible, write “undefined”: 2 4 9A 10 Addition, subtraction, multiplication and division including mixed number representations. Order of operation (up to 3 operations). 1 5 Compute, if possible. Otherwise, write “undefined”: 2 1 3 6 3 2 Compute, if possible. Otherwise, write “undefined”: 2 5 5 7 1 6 Compute, if possible. Otherwise, write “undefined”: 2 1 3 7 2 1 Evaluate, if possible. Otherwise, write “undefined”: 1 (1 ) 3 9 Evaluate, if possible. Otherwise, write “undefined”: 2 1 22 2 1 2 Evaluate, if possible. Otherwise, write “undefined”: (2 3)( ) 3 3 3 1 3 Evaluate, if possible. Otherwise, write “undefined”: 5 2 5 3 4 17 Evaluate, if possible. Otherwise, write “undefined”: 20 17 5 2 2 Evaluate, if possible. Otherwise, write “undefined”: 8 5 3 5 3 Evaluate, if possible. Otherwise, write “undefined”: 2 6 4 2 Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: Evaluate, if possible. Otherwise, write “undefined”: 2 14 7 5 0. 2 20.1 0.01 (0.7 0.3)( 0.02) (0.2)( 5) (0.2)(0.3) 0.9 1.8 10 4.2 (3.9 4.6) 3 2 1 7 1 5 (2) 3 3 22 9 11 Evaluate, if possible. Otherwise, write “undefined”: 1 3 4 3 5 ALGEBRA PART: 36 QUESTIONS EASY CATEGORY: select randomly 8 categories (1 question from each category) 1E Writing phrases (involving only one operation) as algebraic expressions. Write a direct translation of the following phrases as an algebraic expression. DO NOT SIMPLIFY: The sum of a and b Write a direct translation of the following phrases as an algebraic expression. DO NOT SIMPLIFY: x raised to m -th power Write a direct translation of the following phrases as an algebraic expression. DO NOT SIMPLIFY: The product of a and b Write a direct translation of the following phrases as an algebraic expression: The opposite of C Let m represent mass, and c speed of light. Use m and c to write the following phrase as an algebraic expression: the product of mass and the square of speed of light. Use any letter to represent a number and write the following statement as an algebraic expression: Double a number Use any letter to represent a number and write the following statement as an algebraic expression: Two thirds of a number Use any letter to represent a number and write the following statement as an algebraic expression: A number increased by 3 Write a direct translation of the following phrases as an algebraic expression. DO NOT SIMPLIFY: n subtracted from m . Write a direct translation of the following phrases as an algebraic expression. DO NOT SIMPLIFY: The product of a and b . Write a direct translation of the following phrases as an algebraic expression: n divided by m . Do not simplify. 12 Write a direct translation of the following phrases as an algebraic expression: The sum of x , p and y. Write a direct translation of the following phrases as an algebraic expression. DO NOT b SIMPLIFY: raised to the tenth power. a Write a direct translation of the following phrases as an algebraic expression. DO NOT SIMPLIFY: 2ab squared. Write a direct translation of the following phrases as an algebraic expression. DO NOT 2a SIMPLIFY: The opposite of . 7 Write a direct translation of the following phrases as an algebraic expression. DO NOT 2a SIMPLIFY: The opposite of . Do not simplify. 7 Use the letter x to represent a number and write the following phrase as algebraic expression: One-eighth of a number Use the letter x to represent a number and write the following phrase as an algebraic expression: A number multiplied by 3 Use the letter x to represent a number and write the following phrase as an algebraic expression: A number subtracted from 3 Use the letter x to represent a number and write the following phrase as an algebraic expression: The product of AB and a number. Let a be a variable representing the length of the side of a square. Use a to write the following statement as an algebraic expression: Four times the side of a square. Let a be a variable representing the length of the side of a square. Use a to write the following statement as an algebraic expression: The side of a square raised to the second power. Let a be a variable representing the length of the side of a square. Use a to write the following statement as an algebraic expression: The side of a square doubled. John has x dollars in his bank account. How much money will John have in the bank, after he withdraws $100? Give your answer in the form of an algebraic expression. John’s salary is x dollars. He got a new job and his salary doubled. How much does John earn in his new job? Give your answer in the form of an algebraic expression. 13 If a family has x children, and each of those children has 3 children of his own, how many grandchildren are there? Give your answer in the form of an algebraic expression. Let t be a variable representing temperature on a given day. Use the variable to write the following statement as an algebraic expression: The temperature on the given day increased by 5 degrees. 2E Ability to distinguish between an algebraic expression and equation. Determine whether the following examples represent an equation or an algebraic expression. Circle ALL equations. a) 4x 2 b) 4x 2 7 c) A 0 Determine whether the following examples represent an equation or an algebraic expression. Circle ALL equations. 2 2 a) 4x x 2 9x b) 4x 2 7 c) x 3 y 4 x Determine whether the following examples represent an equation or an algebraic expression. Underline the example that represents an equation. a) 4 x 2 5 y b) x 3 2 x c) 4 x 2 5 y 9 Determine whether the following examples represent an equation or an algebraic expression. Circle ALL algebraic expressions. a) 3x 7 y 5 b) x 0 c) 9 x 7 y 3c 8 Determine whether the following examples represent an equation or an algebraic expression. Circle ALL algebraic expressions. a) x 2 2 y 2 0 b) 4 x 5 y z c) x 3E Solving linear equations (all variables on one side, no parentheses involved, no need for collecting terms, no fractions or decimals unless they appear as a solution). Solve the following equation: 98 2 x Solve the following equation: 7 x 2 Solve the following equation: 3x 7 4 14 Solve the following equation: x0 Solve the following equation: 2a 5 Solve the following equation: x 5 1 Solve the following equation: 27 y Solve the following equation: 4 x 8 Solve the following equation: 3 b 2 Solve the following equation: 9 x 8 Solve the following equation: 2 x Solve the following equation: 5 2x 8 Solve the following equation: 7 x 21 0 Solve the following equation: 0 1 x Solve the following equation: 3 6w 2 Solve the following equation: 2x 2 4 Solve the following equation: 2 x 5 Solve the following equation: 2x 3 1 Solve the following equation: 7 3 x 15 4E Solving a linear inequality (all variables on one side, no parentheses involved, no need for collecting terms, no fractions or decimals unless they appear as a solution). Solve the following inequality: 10 2x Solve the following inequality: 9 x 2 Solve the following inequality: x0 Solve the following inequality: x 3 4 Solve the following inequality: 7x 0 Solve the following inequality: 9 x Solve the following inequality: 4 3x Solve the following inequality: 9 2 x Solve the following inequality: x 2 1 Solve the following inequality: 4 x 2 Solve the following inequality: 2x 5 Solve the following inequality: 3 5 x Solve the following inequality: 2 7a 5 Solve the following inequality: 2 3x 1 16 Solve the following inequality: 6 4 x ,,, Solve the following inequality: 7 x 7 Solve the following inequality: 9 2 x Solve the following inequality: 0 3a 1 Solve the following inequality: 8 6x 3 Solve the following inequality: 6 x 1 5E Graphing sets on a number line. Graph the following number set on a number line. Assume that the distance between all marks is the same. x 1 Graph the following number set on a number line. Assume that the distance between all marks is the same. x3 Using inequality symbols, describe the set that is graphed below: Graph the following inequality on the number line. Assume that the distance between all marks is the same. x3 17 Graph the following inequality on the number line. Assume that the distance between all marks is the same. x 1 2 Graph the following inequality on the number line. Assume that the distance between all marks is the same. x 5 2 Graph the following inequality on the number line. Assume that the distance between all marks is the same. x 1 2 Graph the following inequality on the number line. Assume that the distance between all marks is the same. x 2 Graph the following inequality on the number line. Assume that the distance between all marks is the same. x 1 3 18 Graph the following inequality on the number line. Assume that the distance between all marks is the same. x 2 3 Graph the following inequality on the number line. Assume that the distance between all marks is the same. x 2 Graph the following inequality on the number line. Assume that the distance between all marks is the same. x 3 Using inequality symbols, describe the set that is graphed below: Using inequality symbols, describe the set that is graphed below: Using inequality symbols, describe the set that is graphed below: Using inequality symbols, describe the set that is graphed below: 19 6E Checking if a given integer is a solution of a linear equation (inequality) in one variable. Is the number 5 a solution of the following inequality. x 2 4 ? Is the number 5 a solution of the following inequality. x 8 13 ? Is the number 2 a solution of the following inequality. x 3 1 ? For each of the following equations determine if the number 0 is its solution. If 0 is a solution, circle the equation. 1 a) 3x 0 b) 0 x For each of the following equations determine if the number 0 is its solution. If 0 is a solution, circle the equation. a) 3x 22 22 b) x 0 Does x 7 make the statement 2( x 1) x 7 true or false? Does x 2 make the statement x 2 4 true or false? Does x 2 make the statement x 2 4 true or false? Is x 1 a solution of Is x 1 a solution of ( x)( x) 2 xx 2 Is x 2 a solution of 6 x 36 Is x 2 a solution of (6) x 36 Is the number 2 a solution of the following inequality. x 2 . x 2 Is the number 9 is a solution of the following inequality. 3 Is the number 1 a solution of the following inequality. 4 x 3 8 20 Is the number 3 a solution of the following equation. 4x 7 19 . Is the number 100 a solution of the following equation. 432x 4320 . Is the number 1 a solution of the following equation x 1 0 5 x 0 5 2 0 Is the number 0 a solution of the following equation. x Is the number 0 a solution of the following equation. 7E Understanding “<” , “ ” and “>”, “ ” notation. Using it to describe sets of numbers (phrases like “at least”, “at most”, “not less”, “not more”). Circle all of the numbers that satisfy the condition x 2 . 4, 2.1, 2, 0.9, 0, 1 Name three numbers that satisfy the condition: x 1 . Write any inequality that is satisfied by 2 and 3. Write any inequality that is not satisfied by 2 and 3. Write any inequality that is satisfied by 2, but not by 3. Describe the following set of numbers using inequality signs: All numbers x that are positive Describe the following set of numbers using inequality signs: All numbers x that are nonnegative Describe the following set of numbers using inequality signs: All numbers x that are at most equal to 9. Describe the following set of numbers using inequality signs: All numbers x that are at least equal to 9. Describe the following set of numbers using inequality signs: All numbers x that are not less than 9. 21 Describe the following set of numbers using inequality signs: All numbers x that are not more than 1 . Describe the following sets of numbers using inequality signs: All numbers x that are negative. Describe the following sets of numbers using inequality signs: All numbers x that are nonpositive. Describe the following sets of numbers using inequality signs: All numbers x that are at least equal to 6. Describe the following sets of numbers using inequality signs: All numbers x that are at most equal to 6. Describe the following sets of numbers using inequality signs: All numbers x that are not more than 6 Name three numbers that satisfy the condition: x 4 Name three numbers that satisfy the condition: x 3 Name three numbers that satisfy the condition: x 1 2 Name three numbers that satisfy the condition: x 0.5 Find a number that satisfies x 8 but does not satisfy x 100 Circle all numbers that satisfy the following inequality x 3 2, 1, 0, 1, 2, 3, 3, 4, 5 Circle all numbers that do not satisfy the inequality: x 3 2, 1, 0, 1, 2, 3, 3, 4, 5 Circle all numbers that satisfy the inequality: x 2 : 2, 1, 0, 1, 2, 3, 3, 4, 5 8E 22 Rewriting the expression by replacing the variable with its value (only integers) and evaluating it, if possible. One operation needed for the evaluation. Let x 100 . Rewrite the expression replacing the variable with its value and evaluate it, if possible. Otherwise, write “not defined” : 3x Let x 1 . Rewrite the expression replacing the variable with its value and evaluate it, if possible. Otherwise, write “not defined” : x 2 Let x 0 . Rewrite the expression replacing the variable with its value and evaluate it, if 4 possible. Otherwise, write “not defined” : x Let x 100 . Rewrite the expression replacing the variable with its value and evaluate it, if x possible. Otherwise, write “not defined” : 700 Let x 100 . Rewrite the expression replacing the variable with its value and evaluate it, if 0 possible. Otherwise, write “not defined” : x Let x 2 . Rewrite the expression replacing the variable with its value and evaluate, if possible. Otherwise, write “not defined” : 3 x Let x 2 . Rewrite the expression replacing the variable with its value and evaluate, if possible. Otherwise, write “not defined” : x 3 Let x 2 . Rewrite the expression replacing the variable with its value and evaluate, if possible. Otherwise, write “not defined” : x x Rewrite the expression A replacing the variable with its value and then evaluate if A 2 . Substitute x 6 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : x 8 Substitute x 6 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 10 x Substitute x 6 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 4 x Substitute x 6 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : x 6 23 Substitute x 6 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 2 x 6 Substitute x 2 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 2 x Substitute x 2 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 2 x Substitute x 2 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 2 x Substitute x 2 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 5 x 4 Substitute x 2 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 6 x 10 Substitute x 10 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 3x Substitute x 10 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 5x Substitute x 10 in the following expression and then evaluate, if possible. Otherwise, write 200 “not defined” : x Substitute x 10 in the following expression and then evaluate, if possible. Otherwise, write x “not defined” : 2 Substitute x 10 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 50 x Substitute x 10 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : x 4 Substitute x 12 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 1000x Substitute x 12 in the following expression and then evaluate, if possible. Otherwise, write x “not defined” : 6 Substitute x 12 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 5x 24 Substitute x 12 in the following expression and then evaluate, if possible. Otherwise, write 6 “not defined” : x 12 Substitute x 12 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : 24 x Substitute x 12 in the following expression and then evaluate, if possible. Otherwise, write “not defined” : x 2 9E Solving literal equations (variable one is seeking will not appear in the denominator). One operation needed. No simplification needed at the end. Solve for x : Solve for x : Solve for x : Solve for x : x b a xa b ax c xd 0 Solve for the indicated variable: Solve for the indicated variable: Solve for m : Solve for x : b 7c for b a a b 4 for a m x 4 x Y Z Solve for x : xc Solve for x : x z v Solve for x : yx z Solve for x : 4 xm Solve for x : 3 xy 25 Solve for x : y x Solve for the indicated variable: b 7 y z for b Solve for the indicated variable: Solve for m : Solve for x : sa m for a xy 3m x c a 10E If possible, rewriting algebraic expressions with the use of exponential laws (non-negative exponents; only one law used) . Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just rewrite the expression: a 3 a 15 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just rewrite the expression: a 3 a 15 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just rewrite the expression: b 4 2 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just rewrite the expression: a 3 b 4 If possible, eliminate parentheses by applying an exponential law. If it is not possible to simplify, just rewrite the expression: (ab) 7 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just rewrite the expression: a 3b15 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just a7 rewrite the expression: a3 Simplify, using one of the exponential laws, if possible, If not possible, write “not possible” : m 10 2 26 Simplify, using one of the exponential laws, if possible, If not possible, write “not possible” : m3m 4 m If possible, eliminate parentheses by applying an exponential law. If not possible, write “not m possible”: n 7 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just rewrite the expression: xx15 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just s 14 rewrite the expression: s3 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just rewrite the expression: t 15 3 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just rewrite the expression: m 3 m 4 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just rewrite the expression: x 7 x 15 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just rewrite the expression: aa 3 a Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just q 28 rewrite the expression: q 14 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just q 14 rewrite the expression: q 13 Simplify, using one of the exponential laws, if possible, If it is not possible to simplify, just q5 rewrite the expression: q 11E 27 Removing parentheses in multiplication of a monomial by a polynomial (up to 3 terms). Performing numerical operations might be required but no use of exponential laws needed. No fractions, no decimals. Eliminate parentheses in the following expression: 4( x 3 y z ) Eliminate parentheses in the following expression: ( x 2 y) Eliminate parentheses in the following expression: a (b c d ) . Eliminate parentheses in the following expression: 3x 10 2 Eliminate parentheses in the following expression: 2L W Eliminate parentheses in the following expression: R1 x Eliminate parentheses in the following expression: R Eliminate parentheses in the following expression: 3P1 rt Eliminate parentheses in the following expression: ( x 2 7 z )(2) Eliminate parentheses in the following expression: (a a 2 2) Eliminate parentheses in the following expression: (3 xy) . Eliminate parentheses in the following expression: d ( 2a c ) Eliminate parentheses in the following expression: ( x y )( 2 w) Eliminate parentheses in the following expression: ( x 3 x 2 1)4 y Eliminate parentheses in the following expression: (a b g )( d ) . Eliminate parentheses in the following expression: (10 a 3 ) 2 r2 A. 12E Collecting like terms (2 terms only), if possible. No fractions, no decimals. 28 If possible, collect like terms. If not possible, just rewrite the expression 3x 4x If possible, collect like terms. If not possible, just rewrite the expression a 2 6a 2 If possible, collect like terms. If not possible, just rewrite the expression mn m If possible, collect like terms. If not possible, just rewrite the expression xy yx If possible, collect like terms. If not possible, just rewrite the expression 4x 2x If possible, collect like terms. If not possible, just rewrite the expression a 2 a If possible, collect like terms. If not possible, just rewrite the expression y y If possible, collect like terms. If not possible, just rewrite the expression xy yx If possible, collect like terms. If not possible, just rewrite the expression a ab If possible, collect like terms. If not possible, just rewrite the expression st ts If possible, collect like terms. If not possible, just rewrite the expression x 9x If possible, collect like terms. If not possible, just rewrite the expression 4 y 2 9 y 2 If possible, collect like terms. If not possible, just rewrite the expression m 3 m 3 If possible, collect like terms. If not possible, just rewrite the expression xyz 6 zyx If possible, collect like terms. If not possible, just rewrite the expression 3cd 4dcs If possible, collect like terms. If not possible, just rewrite the expression 2ab ba If possible, collect like terms. If not possible, just rewrite the expression 3x 3 4 x 3 y If possible, collect like terms. If not possible, just rewrite the expression 3a 22 6a 22 2 If possible, collect like terms. If not possible, just rewrite the expression st t If possible, collect like terms. If not possible, just rewrite the expression 2mnk 5kmn 13E 29 Factoring an indicated expression (a variable or an integer). No use of exponential laws needed. No instances when “an entire term is factored” and thus “1” becomes a term after factorization. Factor x from the following expression: 3 x 2 xy Factor 4 from the following expression: 4 y 2 8x Factor 5 from the expression: 5 x 5 y Factor 7 from the expression: 21 49a Factor c from the expression: 3c 2ca Factor y from the expression: 8 xy 7 y Factor 2 from the 2lw 2lh 2wh Factor x from the expression: 4 xz yx 5 x Factor x from the following expression: yx zx vx Factor 10 from the following expression: 100 yxz 10v Factor 9 from the expression: 18 x 27 y Factor 2 from the expression: 20 4a 2b Factor c from the expression: 3cd 2c Factor y from the expression: y 14 xy Factor 2 from the expression 2 x 4 yz 12bc Factor x from the expression: 7 xz 4 xy 5 x Factor x from the following expression: xy 2 8 xy Factor 4 from the following expression: 12 xy 2 4 z Factor m from the expression: 5mn 3m Factor s from the expression: st 49s 30 BASIC CATEGORY: 1 question from each category of 23 categories 1B Simplifying an expression by performing numerical operations when possible. (may need to change the order of terms or factors). Performing an operation on fractions or on decimals might be needed. Simplify if possible. 2 x 1 If no numerical operation can be performed, write “not possible”: Simplify if possible. If no numerical operation can be performed, write “not possible”: 2 1 Y 3 6 Simplify if possible. If no numerical operation can be performed, write “not possible”: Simplify if possible. 12 3x If no numerical operation can be performed write 4x 8 “not possible”: Simplify if possible. If no numerical operation can be performed, write “not possible”: (2 3) m Simplify if possible. If no numerical operation can be performed, write “not possible”: 2 3m Simplify if possible. If no numerical operation can be performed, write “not possible”: 2 x ( 3) y Simplify if possible. If no numerical operation can be performed, write “not possible”: ( x) Simplify if possible. 0.2 x 0.03 If no numerical operation can be performed, write “not possible”: Simplify if possible. If no numerical operation can be performed, write “not possible”: Simplify if possible. 4a(8) If no numerical operation can be performed, write x 1 “not possible”: Simplify if possible. If no numerical operation can be performed, write “not possible”: 8a 4 31 Simplify if possible. If no numerical operation can be performed, write “not possible”: 1 1 x 2 2 Simplify if possible. If no numerical operation can be performed, write “not possible”: 3bd 9ac Simplify if possible. If no numerical operation can be performed, write “not possible”: 2 x ( 3) y Simplify if possible. 2 4 x 5 If no numerical operation can be performed, write “not possible”: Simplify if possible. If no numerical operation can be performed, write “not possible”: 12 y 2 xz Simplify if possible. If no numerical operation can be performed, write “not possible”: 1 2x 2 2 Simplify if possible. If no numerical operation can be performed, write “not possible”: x 10 5 Simplify if possible. If no numerical operation can be performed, write “not possible”: 0.1x(10 y 2 ) Simplify if possible. If no numerical operation can be performed, write “not possible”: 0.5(0.3m) Simplify if possible. If no numerical operation can be performed, write “not possible”: 0.5 0.6 m Simplify if possible. If no numerical operation can be performed, write “not possible”: 0 .4 y 0.02 2B 32 Simplifying algebraic expressions by applying exponential laws (only nonnegative exponents; one variable, up to 2 types of operations. Performing an operation on fractions or on decimals might be needed. Simplify Simplify: 2( x 2 ) 3 x a3 aa 2 Perform the indicated operation and simplify: Perform the indicated operation and simplify: (3x 5 )( 4 x 2 ) 3x 4 6x 2 Perform the indicated operation and simplify: 3x(4 x 2 ) Perform the indicated operation and simplify: (2 x 6 ) 3 Perform the indicated operation and simplify: Perform the indicated operation and simplify: 8x 5 6x 2 2x3 x5 x4 Perform the indicated operation and simplify: a2 2 a a4 Perform the indicated operation and simplify: 1 7 v v Perform the indicated operation and simplify: (3x 2 ) 3 x4 Perform the indicated operation and simplify: 3x 5 (3 x ) 2 2 Simplify (3aa 2 ) 2 Simplify: a3 aa 2 Perform the indicated operation and simplify: (4 x) 3 x 33 (a) 3 2a 2 Perform the indicated operation and simplify: Perform the indicated operation and simplify: (x 2 )5 x 7 Perform the indicated operation and simplify: (2 x 6 ) 3 Perform the indicated operation and simplify: (3x 2 )(4 x 5 x) Perform the indicated operation and simplify: (3x)(2)( x 3 ) 3B The Distributive Law: removing parentheses in multiplication of a monomial by a polynomial with up to 3 terms. No collecting terms needed at the end but numerical operations or application of exponential laws needed in the process. Performing an operation on fractions or on decimals might be needed. Use the Distributive Law to eliminate parentheses in the following expression. Simplify, if possible. x(2 x 2 ) Use the Distributive Law to eliminate parentheses in the following expression. Simplify, if possible. a(a a 3 a 4 ) . Use the Distributive Law to eliminate parentheses in the following expression. Simplify, if 1 possible. 3 x 2 x 2 Use the Distributive Law to eliminate parentheses in the following expression. Simplify, if possible. 2( x 3 y 4 z ) Remove parentheses and simplify: ( x 2 2 x) x 4 2 Remove parentheses and simplify: (3c 33d ) 3 Remove parentheses and simplify: a(a 2 ab ab 2 ) Remove parentheses and simplify: (2 x 3 y 3 xy 4 ) x 7 34 2 Remove parentheses and simplify: (a b)( a 3 ) 5 Remove parentheses and simplify: 1 x( 4 x 5 8) 4 Remove parentheses and simplify: 3 y(6 y 4 8 y 3 ) Remove parentheses and simplify: x 2 ( x 5 2 x 3 ) Remove parentheses and simplify: 2 y 2 (7 y y 2 ) Remove parentheses and simplify: a 4 (9a 2 a 2 b) Remove parentheses and simplify: ab(a 2 b ab 5 ) Remove parentheses and simplify: 0.2c(0.3c 100c 6 ) Remove parentheses and simplify: x10 (4 x 5 xy) Remove parentheses and simplify: mn 2 (m n) Remove parentheses and simplify: x 2x5 4x 2 Remove parentheses and simplify: 0.7(0.3 n) Remove parentheses and simplify: (2.3x 0.06)(100) Remove parentheses and simplify: 5(0.06 x 2 y ) Remove parentheses and simplify: 0.6(0.2n 4) 4B The Distributive Law: removing parentheses in multiplication of a binomial by a binomial. No collecting terms needed at the end but numerical operations or application of exponential laws needed in the process (no operations on fractions or decimals). 35 Eliminate parentheses in the following expression. Simplify: (a b)( 2a 3) Eliminate parentheses in the following expression. Simplify: (2b c)(c c 5 ) Eliminate parentheses in the following expression. Simplify ( x y)( zx 2 w) Eliminate parentheses in the following expression. Simplify (a 3b)(10 a 3 ) Eliminate parentheses in the following expression. Simplify (a g )(ca dg 2 ) Eliminate parentheses in the following expression. Simplify: ( x 3 x 2 )( yx 1) Eliminate parentheses in the following expression. Simplify: ( yx 1)( y 1) Eliminate parentheses in the following expression. Simplify (2ab 5 1)(1 ba) Eliminate parentheses in the following expression. Simplify (4 x 8)( 2 yx 3 3x 9 ) Eliminate parentheses in the following expression. Simplify (m 2 m)(3m 5 1) Eliminate parentheses in the following expression. Simplify: (4a b)(3ab 2 3a 2 b 3 ) Eliminate parentheses in the following expression. Simplify: (b 4 c)(c cb 5 ) Eliminate parentheses in the following expression. Simplify (3x 2 y xy)(2 x y 3 ) Eliminate parentheses in the following expression. Simplify (2b a 3 )(2a b) Eliminate parentheses in the following expression. Simplify ( x y 2 )( y 3) Eliminate parentheses in the following expression. Simplify: ( x 2 1)(a 2 x) Eliminate parentheses in the following expression. Simplify: (3m n)( x 2my) Eliminate parentheses in the following expression. Simplify ( x 3 y)(3x 2 2 y) Eliminate parentheses in the following expression. Simplify (3x xy)(1 3x) Eliminate parentheses in the following expression. Simplify (2 z )( z v) 36 5B Factorization of an indicated monomial (factorization of “an entire term” is and thus getting “1” as a term after factorization possible). No operations on fractions or decimals. Factor x 2 from the following expression: 3x 5 2 x 3 x 2 Factor 4a from the following expression: 4a 8 xa 4 Factor x 2 from the following expression: 3x 5 yx 2 Factor xy 2 from the following expression: 4 x 2 y 2 xy 4 3x 3 y 3 Factor 2 xy from the following expression: 2 xy 2 xy2 4 x 2 y 2 Factor a 3b 3 from the following expression: a 3b 4 5b 3 a 7 a 3b 3 Factor 7 xy from the following expression: 7 x 5 y 3 14 x 3 y 2 21xy Factor 7a 3 b 3 from the following expression: 14b 3 a 7 7a 3b 3 Factor 4ac 3 from the following expression: 16ac 3 8ac 7 12ac 8 d Factor 7ab 2 from the following expression: 35ab 3 14a 2 b 4 7ab 2 Factor 2 y from the following expression: 2 x 2 y 4 y 4 Factor x 2 from the following expression: 2 x 2 3 x 4 Factor 2 x 2 y from the following expression: 6 x 2 y 14 y 7 x 3 Factor 3abc 2 from the following expression: 3abc 2 12a 6 bc 5 6a 2 b 2 c 2 Factor z 3 from the following expression: 3z 5 z 4 2 z 3 Factor x 2 from the following expression: 2 x 3 x 2 Factor 3ay from the following expression: 3a 3 y 3x 2 ay 6a 4 y 2 37 Factor 3ab from the following expression: 6ab 5 3ba Factor a 2 b from the following expression: a 3b a 2 b 4a 7 b 2 Factor 3z 2 y 3 from the following expression: 3z 4 y 6 9 y 4 z 2 6B Simplification of rational expressions. Factorization of a numerator OR a denominator (but not both) needed. a 2ab a xy Simplify, if possible. If not possible, write “not possible”: 2 x y 3xy 2 Simplify, if possible. If not possible, write “not possible”: Simplify, if possible. If not possible, write “not possible”: a 4d 4d a Simplify, if possible. If not possible, write “not possible”: 2 2x 2 y Simplify, if possible. If not possible, write “not possible”: xy xz 3x Simplify, if possible. If not possible, write “not possible”: 3x 3x 2 6x 4 x 5x 2 Simplify, if possible. If not possible, write “not possible”: x m 3n Simplify, if possible. If not possible, write “not possible”: 3n m Simplify, if possible. If not possible, write “not possible”: a 3b 4b 4 a 4 a Simplify, if possible. If not possible, write “not possible”: 3x 3x 9 Simplify, if possible. If not possible, write “not possible”: 3x 3x 9 x 2 38 Simplify, if possible. If not possible, write “not possible”: x2 2x 2 x5 Simplify, if possible. If not possible, write “not possible”: abc ab Simplify, if possible. If not possible, write “not possible”: xy xy xy 2 Simplify, if possible. If not possible, write “not possible”: 12 x 4 3x Simplify, if possible. If not possible, write “not possible”: u 2v s s 2v u Simplify, if possible. If not possible, write “not possible”: 4uv 2 2vu 2uv Simplify, if possible. If not possible, write “not possible”: xy xy 2 3 yx Simplify, if possible. If not possible, write “not possible”: 4 x 12 y 8 z 4 Simplify, if possible. If not possible, write “not possible”: 8z 4x 8z 7B Understanding what it means that a number(s) is a solution of a given equation (“relatively” complicated evaluation needed). Performing an operation on fractions or on decimals might be needed. Ability to rewrite the expression and evaluating it, if possible (one instance of a variable). If evaluation is not possible, recognizing that it is not possible to evaluate. Performing an operation on fractions or on decimals might be needed. Circle all of the numbers below that are solutions of the equation x 3 6 x . 39 x2 x 2 Determine if x 1, y 2 is a solution of the following equation. Show your work. 3x 2 y 1. Determine if x 1, y 2 is a solution of the following equation. Show your work. x y 3 Determine if x 1, y 2 is a solution of the following equation. Show your work. y 2 x Does a 1 , b 2 make the following statement true or false? Show your work. 2 ab 1 Does a 1 , b 2 make the following statement true or false? Show your work. 2 ab 1 1 Does a , b 2 make the following statement true or false? Show your work. 2 a 1 b 4 1 Does a , b 2 make the following statement true or false? Show your work. 2 3 ab 2 Is x 1, y 0 a solution of 2 xy 2 ? Show your work. Is x 1, y 0 a solution of 2 xy 0 ? Show your work. Is x 1, y 0 a solution of y x 1 ? Show your work. Is x 1, y 0 a solution of x 0 ? Show your work. y 1 , y 2 a solution of 2 x 4 ? Show your work. y Is x 40 Is x 1, y 0 a solution of y 4 ? Show your work. x Is x 1, y 0 a solution of y x 3 ? Show your work. 2 Is x 1, y 0 a solution of xy y 2 x ? Show your work. Let x 4 . Rewrite the expression replacing the variable with its value and evaluate: 2x Let x 4 . Rewrite the expression replacing the variable with its value and evaluate: (2) x Let x 8 . Rewrite the expression replacing the variable with its value and evaluate: ( x) 2 Let x 100 . Rewrite the expression replacing the variable with its value and evaluate: Substitute x x2 2 5 x in the following expression and then evaluate: 3 3 2 in the following expression and then evaluate: 3 2 Substitute x in the following expression and then evaluate: 3 Substitute x x 1 5 x 2 7 Substitute x 2 in the following expression and then evaluate: 3 5 x 12 Substitute x 2 in the following expression and then evaluate: 3 2 x Substitute x 2 in the following expression and then evaluate: 3 x 3 Substitute x 3 in the following expression and then evaluate: 5 3 x 10 Substitute x 3 in the following expression and then evaluate: 5 1 x 7 41 Substitute x 3 in the following expression and then evaluate: 5 1 2 x 5 Substitute x 3 in the following expression and then evaluate: 5 1 1 x 4 Substitute x 3 in the following expression and then evaluate: 5 x3 Substitute x 2 in the following expression and then evaluate: 7 2x Substitute x 2 in the following expression and then evaluate: 7 7x Substitute x 2 in the following expression and then evaluate: 7 Substitute x 2 in the following expression and then evaluate: 7 5 x 28 Substitute x 2 in the following expression and then evaluate: 7 5 x Substitute x 2 in the following expression and then evaluate: 7 x 2 Substitute x 3 in the following expression and then evaluate: 4 1 2 14 x 3 x2 Substitute x 3 in the following expression and then evaluate: 4 3 Substitute x in the following expression and then evaluate: 4 4 x 3 x 2 1 3 3 in the following expression and then evaluate: 4 3 Substitute x in the following expression and then evaluate: 4 1 1 x 8 x 3 Substitute x Substitute x 3 in the following expression and then evaluate: 4 0 x 42 Substitute x 0.2 in the following expression and then evaluate: x 3.21 Substitute x 0.2 in the following expression and then evaluate: 35.01 x Substitute x 0.2 in the following expression and then evaluate: x 4 Substitute x 0.2 in the following expression and then evaluate: 40x Substitute x 0.2 in the following expression and then evaluate: 0.3x Substitute x 0.2 in the following expression and then evaluate: x 0.04 Substitute x 0.6 in the following expression and then evaluate: x 4.5 Substitute x 0.6 in the following expression and then evaluate: 2.7 x Substitute x 0.6 in the following expression and then evaluate: 1 .2 x Substitute x 0.6 in the following expression and then evaluate: 600x Substitute x 0.6 in the following expression and then evaluate: 0.001x Substitute x 0.6 in the following expression and then evaluate: x3 Substitute x 1.5 in the following expression and then evaluate: x 0.08 Substitute x 1.5 in the following expression and then evaluate: 3 x 0.4 Substitute x 1.5 in the following expression and then evaluate: x 0.15 Substitute x 1.5 in the following expression and then evaluate: 0.2x Substitute x 1.5 in the following expression and then evaluate: 30 x 8B 43 Recognizing equivalent expressions by applying the following a) the order of terms of an expression can be changed b) the order of factors of an expression can be changed c) b c bc a a a d) ab b 1 a ab c c c a b e) a a b b Circle all expressions that are equal to 2 x 5 y 4 z : 2x 4z 5 y 5 y 4z 2x 5 y 2x 4z Rewrite the expression yx(a b) in two equivalent forms by multiplying its factors in a different order. Rewrite the following expression in its equivalent form as a single fraction: 6 a 11 Rewrite the following expression in its equivalent form as a single fraction: 2 x y Rewrite the following expression in its equivalent form as a single fraction: 1 2 (a 1) 3 Rewrite the following expression in its equivalent form as a single fraction: a 2b x2 x2 Fill in the blanks to make a true statement: 3x x _____ y Fill in the blanks to make a true statement: 3x 1 ____ y y ab 1 _____ y y 2b Circle all expressions that are equivalent to c b2 2 b 1 b2 , ( 2 b) , , c c c c c Fill in the blanks to make a true statement: Circle all expressions that are equivalent to m m m m(1) , , , 1 1 1 m 44 Circle all expressions that are equivalent to m n : m( n ) , n m, n m, nm , Circle all expressions that are equivalent to 2 xy , 9 2 xy , 9 2 xy 1 , 9 Circle all expressions that are equivalent to 1 xy 2 , 9 xy 2, 9 2xy 9 2xy 9 2x y 9 Circle all expressions that are equivalent to a c b d . a b c d, a b c d, d b c a, Circle all expressions that are equivalent to a , b 1 a , b 2a , 2b a . b 1 b , a Circle all expressions that are equivalent to acbd a , b a b a . b Circle all expressions that are equivalent to abc cba b(a)c a bc 1 Circle all expressions that are equivalent to abc bca ab(c) ab c 1 Circle all expressions that are equivalent to 5x . 6 45 5 5 x, 6 x , 6 1 5x 6 Circle all expressions that are equivalent to 5 x , 6 5 , 6x 5x . 6 10 x , 12 Circle all expressions that are equivalent to x , 2x , 2 x, 2 x2 . Circle all expressions that are equivalent to 4x x x2, , 8 2 x2 . Circle all expressions that are equivalent to 11a , 8a 3 , 3 a 8 , 3 8a . Circle all expressions that are equivalent to b 3a , 6 1 (3a b), 6 b 3a , 6 (3a b) mn , 3 3a b 6 1 6 Circle all expressions that are equivalent to n m , 3 3 3a b . 6 3a b, 6 Circle all expressions that are equivalent to 3a b , 6 6 11 a , m n . 3 3 1 1 m n, 3 3 By placing the minus sign differently, write the following expression in two additional 2a equivalent ways. b 46 By placing the minus sign differently, write the following expression in two additional 2ac equivalent ways. 2d m 7n Rewrite the following expression as of a single fraction: 4 4 7m n 2 t t 5m 2n 2 Rewrite the following expression as of a single fraction: 4c 2 4c 2 Rewrite the following expression as of a single fraction: Rewrite the following expression as of a single fraction: m 3 t s 1 s 1 s 1 Using the rule for multiplication of fractions, rewrite the following expression in its equivalent m form as a single fraction: 3 n Using the rule for multiplication of fractions, rewrite the following expression in its equivalent b form as a single fraction: a 4 Using the rule for multiplication of fractions, rewrite the following expression in its equivalent b form as a single fraction: a 4 Using the rule for multiplication of fractions, rewrite the following expression in its equivalent t form as a single fraction: ( s 4) n Using the rule for multiplication of fractions, rewrite the following expression in its equivalent a form as a single fraction: 4 n 1 Write the following expressions in its equivalent form as a single fraction: 2 4 x y 3 3 Write the following expressions in its equivalent form as a single fraction: 2 1 x y 3 3 1 y Write the following expressions in its equivalent form as a single fraction: 2 x 7 3 3 Write the following expressions in its equivalent form as a single fraction: 3 1 x y t t 47 Write the following expressions in its equivalent form as a single fraction: Write the following expression in its equivalent form as a single fraction: x y 3 t t 1 n 2 k t k t 2 9B Expanding exponential expression; Rewriting expressions using exponential notation whenever possible. Understanding that the exponent pertains only to the closest expression. Evaluation of exponential expressions raised to the zero-th power. Identifying bases, exponents and numerical coefficients of exponential expressions. Expand (i.e. write without using exponential notation). Do not simplify. a(bc) 2 Expand (i.e. write without using exponential notation). Do not simplify. 5A 3 Rewrite using exponential notation whenever it is possible: Rewrite using exponential notation whenever it is possible: Rewrite using exponential notation whenever it is possible: Rewrite using exponential notation whenever it is possible: Evaluate 7mmm aaaa aa zzz zzzz ( z )( z )( z ) zzz (2 x) 0 Evaluate 2x 0 Evaluate 4 (8 m 7 ) 0 3a 27 In the following expression, identify base, exponent and numerical coefficient: Base___________ exponent__________ numerical coefficient_______________ In the following expression, identify base, exponent and numerical coefficient: yz x 9 48 Base___________ exponent__________ numerical coefficient_______________ In the following expression, identify base, exponent and numerical coefficient: x 4 Base___________ exponent__________ numerical coefficient_______________ 2x3 In the following expression, identify base, exponent and numerical coefficient: 3 Base___________ exponent__________ numerical coefficient_______________ In the following expression, identify base, exponent and numerical coefficient: (a bc) 2 Base___________ exponent__________ numerical coefficient_______________ x In the following expression, identify base, exponent and numerical coefficient: y Base___________ exponent__________ numerical coefficient_______________ m ( x y) 7 In the following expression, identify base, exponent and numerical coefficient: 4 Base___________ exponent__________ numerical coefficient_______________ 3 3x z 4 w Base___________ exponent__________ numerical coefficient_______________ 7 In the following expression, identify base, exponent and numerical coefficient: (ab) 5 2 Base___________ exponent__________ numerical coefficient_______________ In the following expression, identify base, exponent and numerical coefficient: Write using exponential notation whenever it is possible: 3a 3a 3a 3a Write using exponential notation whenever it is possible: xyxyxxx Write using exponential notation whenever it is possible: a aaaa Write using exponential notation whenever it is possible: xxy yyx Write using exponential notation whenever it is possible: (a b)( a b)( a b) 49 Write using exponential notation whenever it is possible: (2t 3 )( 2t 3 )( 2t 3 )( 2t 3 ) Write using exponential notation whenever it is possible: m nn m Write using exponential notation whenever it is possible: kkk k n x Write using exponential notation whenever it is possible: x x x 2 Write using exponential notation whenever it is possible: (3 x)( x 3)(3 x) 2a a 2a Write using exponential notation whenever it is possible: 2 b b b 3 3 3 Write using exponential notation whenever it is possible: x x x 1 Write using exponential notation whenever it is possible: ( w 2v)( 2v w)( 2v w) x y x y Write using exponential notation whenever it is possible: m m m Write using exponential notation whenever it is possible: (m n)( m n)m n Write using exponential notation whenever it is possible: (m p n)( p m n)( n m p) Write the following expression without using exponential notation: (m) 3 Write the following expression without using exponential notation: m 3 Write the following expression without using exponential notation: (xy) 3 Write the following expression without using exponential notation: 2a 3 Write the following expression without using exponential notation: (a b) 2 Write the following expression without using exponential notation: a b 2 Simplify by raising to the indicated power: (3x) 0 Simplify by raising to the indicated power: 3x 0 50 Simplify by raising to the indicated power: 3 0 x Simplify by raising to the indicated power: a(b c) 0 Simplify by raising to the indicated power: abc 0 Simplify by raising to the indicated power: (abc) 0 Simplify by raising to the indicated power: ab c 0 Simplify by raising to the indicated power: a 0b 0 c 0 10B Determining if rational expressions can be simplified and simplifying them. Factorization not needed. Simplify, if possible. If not possible, write “not possible”: 3 xy 9 yx Simplify, if possible. If not possible, write “not possible”: a2 b2a2 Simplify, if possible. If not possible, write “not possible”: a 2 (b c) 2a Simplify, if possible. If not possible, write “not possible”: v 4z 4z Simplify, if possible. If not possible, write “not possible”: 12 xy 9x Simplify, if possible. If not possible, write “not possible”: 2abc 8ab Simplify, if possible. If not possible, write “not possible”: 2mn 2 4n 2 Simplify, if possible. If not possible, write “not possible”: 5 xy 4 20 y 3 51 Simplify, if possible. If not possible, write “not possible”: 15 x(a b) 25 x Simplify, if possible. If not possible, write “not possible”: a 2 (b c) bc Simplify, if possible. If not possible, write “not possible”: bc(b e) b Simplify, if possible. If not possible, write “not possible”: 4 x(5 x 2 ) x Simplify, if possible. If not possible, write “not possible”: ab 7( a b) Simplify, if possible. If not possible, write “not possible”: 2x 3 2x Simplify, if possible. If not possible, write “not possible”: 4x 2 y 3 12 x Simplify, if possible. If not possible, write “not possible”: a 3b 3b a Simplify, if possible. If not possible, write “not possible”: ab c ab Simplify, if possible. If not possible, write “not possible”: 2( x 3 y ) 3y x Simplify, if possible. If not possible, write “not possible”: 2x 2 z 7 xyz Simplify, if possible. If not possible, write “not possible”: 3 9(mn k ) 11B Collecting like terms: more than one type of like terms in the expression (fractions and decimals used but not as coefficients of variables). Simplify by collecting like terms: 3x a 4x a 52 Simplify by collecting like terms: 7 y 5 12 y y 6 Simplify by collecting like terms: a 2 b 3 4b 2 a 3 6a 2 b 3 Simplify by collecting like terms: 0.2 x x 0.8 Simplify by collecting like terms: 2 1 xy y y yx 3 3 Simplify by collecting like terms: 3 x 8 y 8 x 2 x Simplify by collecting like terms: 5a 3b 2b 7a Simplify by collecting like terms: 2ab 4ba 2 3ab 1 Simplify by collecting like terms: 3 j2 4 j2 2 j Simplify by collecting like terms: a 2 b 3 4b 2 a 3 6a 2 b 3 Simplify by collecting like terms: 0.2 2z 5z z 0.4 Simplify by collecting like terms: x3 x4 x3 Simplify by collecting like terms: Simplify by collecting like terms: Simplify by collecting like terms: 1 2x 1 2 3y 7x x 4 y 2 2 7m 4 5 x Simplify by collecting like terms: ab ba 0.7 0.9 ba Simplify by collecting like terms: x 4 y 2 x 3 yx 4 5x Simplify by collecting like terms: 2 1 cd dc d c 3 3 Simplify by collecting like terms: 2 a 2a 2.3 Simplify by collecting like terms: 2x y x 2 y 12B 53 Solving linear equations with a variable on both sides. No fractions or parentheses involved. “No solution” or “all real numbers” as a solution possible. No operations on decimals needed. Solve the following equation: 3x 5 4x Solve the following equation: 4x 5 15 x Solve the following equation: x 8 x 9 Solve the following equation: 2 a a Solve the following equation: 3j 4j 2j Solve the following equation: 2 7m 4 m Solve the following equation: z 1 z z Solve the following equation: 7 x 8x Solve the following equation: 4x x 12 Solve the following equation: 6x 15x Solve the following equation: 3 7x x Solve the following equation: 3x 5 4x Solve the following equation: 4x 2 3x Solve the following equation: 3x 7 1 4x Solve the following equation: 4x 3 12 x Solve the following equation: 6 6x 2x 5 Solve the following equation: 30 x 9 21 x Solve the following equation: x 3 3x 2x Solve the following equation: x 4 x 2 Solve the following equation: 2x 3 x 4 54 13B Solving linear inequalities (no parentheses, no fractions, no decimals involved; if a variables is only on one side its coefficient will be negative). “No solution” or “all real numbers” as a solution possible. Solve the following inequality: x x Solve the following inequality: x x Solve the following inequality: x4 Solve the following inequality: x 2 1 Solve the following inequality: 2 3x 14 Solve the following inequality: 8 2 3x Solve the following inequality: 7 x 1 Solve the following inequality: 4x 1 2x 5 Solve the following inequality: 2 6x 8 Solve the following inequality: 1 7 x 5 Solve the following inequality: x 1 2 Solve the following inequality: x 8 x 9 Solve the following inequality: 3 a 5 4 Solve the following inequality: 3x 1 3x 11 Solve the following inequality: 2 a 4a 5a 2 Solve the following inequality: 5x 2 3x 2 Solve the following inequality: 3 2a 6a Solve the following inequality: 3 2 5x Solve the following inequality: 2x 5 6x 16 Solve the following inequality: 3x 18 55 14B “The difference of squares” factorization (formula not given). Factor the following expression: 36 x 2 1 Factor the following expression: x 2 0.09 Factor the following expression: m 4 n 2 Factor the following expression: 1 9x 2 4 Factor the following expression: y 2 100a 2 Factor the following expression: x 2 y 2 25 Factor the following expression: 25s 2 64t 2 Factor the following expression: a2 0.01 b2 Factor the following expression: m 2 81n 2 Factor the following expression: (2a) 2 9a 2 Factor the following expression: 4 x 2 36 y 2 Factor the following expression: x2 1 36 Factor the following expression: 1 m 2 n 2 Factor the following expression: x 4 9 25 Factor the following expression: 9m 4 1 16 Factor the following expression: 25 x 2 (7 y ) 2 Factor the following expression: x 6 16 56 Factor the following expression: 0.04 s 4 Factor the following expression: 4 s 12 Factor the following expression: 25 x 10 4 15B Rewriting algebraic expression in terms of another variable. Only a direct substitution needed; “relatively easy” simplification. Rewrite the expression 3a in terms of x , given that a 2 x (i.e. substitute 2x for a ) Simplify. 2 Express 2 P in terms of m , when P m (i. e. substitute m for P ). Write your final answer without parentheses. Express 2 P in terms of m , when P m 2 m (i. e. substitute m 2 m for P ). Write your final answer without parentheses. Express A3 in terms of x , when A 2x (i.e. substitute 2x for A). Write your answer without parentheses. Simplify. Express A3 in terms of x , when A x (i.e. substitute x for A). Write your answer without parentheses. Simplify. Rewrite the expression a 1 in terms of x , if it is given that a 3x (i.e. substitute 3x for a). 3x 1 Simplify. Express A3 in terms of x , when A x 7 (i.e. substitute x 7 for A). Write your answer without parentheses. Simplify. Express A3 in terms of x , when A x 3 (i.e. substitute x 3 for A). Write your answer without parentheses. Simplify. Write the expression a 2b in terms of x , if a 5x and b x (i.e. substitute 5x for a and x for b ). Eliminate parentheses and simplify. 57 Write the expression a 2 4a in terms of x , if a x 2 (i.e. substitute x 2 for a). Eliminate parentheses and simplify. x y in terms of a and b if x a b , y a b (i.e. substitute a b for x and x y a b for y and simplify). . Eliminate parentheses and simplify. Express Express the following expression in terms of x (i. e. substitute x for y). Simplify the new expression: x y if y x x y Let P x 2, Q 4 x 1 . Express P Q in terms of x (i.e. substitute x 2 for P and 4x 1 for Q ). Write your final answer without parentheses, in a simplified form. Let P x 2, Q 4 x 1 . Express P Q in terms of x (i.e. substitute x 2 for P and 4x 1 for Q ). Write your final answer without parentheses, in a simplified form. Let P x 2, Q 4 . Express PQ in terms of x (i.e. substitute x 2 for P and 4 for Q ). Write your final answer without parentheses, in a simplified form. Let Q 4 x 1 . Express Q 2 in terms of x (i.e. substitute 4x 1 for Q ). Write your final answer without parentheses, in a simplified form. Let A x 2 , B 4x . Express A B in terms of x (i.e. substitute x 2 for A and 4x for B). Write the expression without parentheses. Simplify. Let A x 2 , B 4x . Express AB in terms of x (i.e. substitute x 2 for A and 4x for B). Write the expression without parentheses. Simplify. Express A 2 AB in terms of x , if A x, B x (i.e. substitute x for A and x for B). Write your answer without parentheses. Simplify. Let P x 3 and R 2x 2 . Write the expression PR in terms of x (i.e. substitute x 3 for P and 2x 2 for R). Simplify. Let P x 3 and R 2x 2 . Express the expression P in terms of x (i.e. substitute x 3 for P R and 2x 2 for R). Simplify. 58 Rewrite the expression a 2 b 2 in terms of x , if a 5x and b 1 (i.e. substitute 5x for a and 1 for b). Write the new expression without parentheses. Simplify. Rewrite the expression a 2 b 2 in terms of x , if a x and b x (i.e. substitute x for a and x for b). Write the new expression without parentheses. Simplify. Rewrite the expression a 2 b 2 in terms of a, if a b. (i.e. substitute b for a). Simplify. Rewrite the expression a 2 b in terms of x , if a x 3 , b x 4 (i.e. substitute x 3 for a and x 4 for b). Write the new expression without parentheses. Simplify. Rewrite the expression a 2 b in terms of a, if a b. (i.e. substitute b for a). Simplify. Let C 3x, D x . Express the expression CD in terms of x (i.e. substitute 3x for C and 2 x for D ). Simplify. 2 Let C 3x, D x . Write the expression C D in terms of x (i.e. substitute 3x for C and 2 x for D ). Simplify. 2 Rewrite the expression a 2 2ab b 2 in terms of x , if x for b. Simplify. a 1, b x , i.e. substitute 1 for a and Rewrite the expression a 2 2ab b 2 in terms of x , if a 3x, b 2 x , i.e. substitute 3x for a and 2x for b. Simplify. Rewrite the expression a 2 2ab b 2 in terms of x , if a and x for b. Simplify. x x , b x , i.e. substitute for a 2 2 Rewrite the expression a 2 2ab b 2 in terms of x , if a b x , i.e. substitute x for a and for b. Simplify Express 2 x y in terms of s , if x s 3 , y 2 s , i.e. substitute s 3 for x and 2s for y . Simplify, if possible. 59 Express x 2 y in terms of s , if x s 3 , y 2 s . i.e. substitute s 3 for x and 2s for y . Simplify, if possible. Express 2 x y 3 in terms of s , if x s 3 , y 2 s . i.e. substitute s 3 for x and 2s for y . Simplify, if possible. xy 2 in terms of s , if x s 3 , y 2 s . i.e. substitute s 3 for x and 2s for y . Simplify, 4 if possible. Express Express ab in terms of x , if a 5x 126 5x 126 , i.e. substitute for a and for b . , b 126 5 126 5 Simplify. x t x t x t 2 y , i.e. substitute y, for Express in terms of y, if y , and 2 y for . z z z z z z Simplify. 2 2 x t x x t 2 y , i.e. substitute 3y, for Express in terms of y, if 3y, and 2 y for z z z z z t . Simplify. z 16B Simplifying algebraic expressions by removing parentheses first, and then collecting like terms. Performing an operation on fractions or on decimals might be needed (no expressions of the form (a b) 2 ) Eliminate parentheses and simplify by collecting like terms: 2(t 5) 4t Eliminate parentheses and simplify by collecting like terms: 3( x 5 2) (4 x 5 ) Eliminate parentheses and simplify by collecting like terms: (q 6)( 8) 21q Eliminate parentheses and simplify by collecting like terms: (a 3)( a 4) Eliminate parentheses and simplify by collecting like terms: 2x ( 2 x 1) 5 Eliminate parentheses and simplify by collecting like terms: 2(a 3) (a 4) 2x ( 2 x 1) Eliminate parentheses and simplify by collecting like terms: 5 Eliminate parentheses and simplify by collecting like terms: 2 x 5 y 4(3x y ) 60 Eliminate parentheses and simplify by collecting like terms: 3xy 7 yx ( xy 3) Eliminate parentheses and simplify by collecting like terms: (6 x 4 3x 3 1) (3x 3 3x 2 3) Eliminate parentheses and simplify by collecting like terms: a 0.1(2 3a) Eliminate parentheses and simplify by collecting like terms: 3cb abc 2(a bc) 1 1 Eliminate parentheses and simplify by collecting like terms: a b (a 2b) 3 3 2 1 Eliminate parentheses and simplify by collecting like terms: 6 ( d a ) a d 3 2 2 Eliminate parentheses and simplify by collecting like terms: 3a 1 (4a 2 2) 4 Eliminate parentheses and simplify by collecting like terms: 3a 9b (4a b) 5 Eliminate parentheses and simplify by collecting like terms: (2q 6) 2q 5q Eliminate parentheses and simplify by collecting like terms: ( x 1)( x) x Eliminate parentheses and simplify by collecting like terms: ( x 1) x Eliminate parentheses and simplify by collecting like terms: 2(t 3) t 1 Eliminate parentheses and simplify by collecting like terms: 0.2 y 3( y 0.3) Eliminate parentheses and simplify by collecting like terms: (6a 2) 12a Eliminate parentheses and simplify by collecting like terms: 4a 7a 2(8a 1) 17B Collecting like terms (if possible) involving operations on fractions, or decimals. If possible, collect like terms. If not possible, write “not possible” 0.03x 0.4 x 2 If possible, collect like terms. If not possible, write “not possible” 1 2 3 2 a a 2 7 If possible, collect like terms. If not possible, write “not possible” 0.5mn 0.7m If possible, collect like terms. If not possible, write “not possible” ab 1 ba 3 6 61 m m 4 6 If possible, collect like terms. If not possible, write “not possible”: If possible, collect like terms. If not possible, write “not possible”: 2m 2m 38 38 If possible, collect like terms. If not possible, write “not possible”: 0.7abc 0.08cba If possible, collect like terms. If not possible, write “not possible”: ab 8.2ba 2 3s 3 If possible, collect like terms. If not possible, write “not possible”: s 3 11 22 If possible, collect like terms. If not possible, write “not possible”: 1 2 x x 3 7 If possible, collect like terms. If not possible, write “not possible” : 0.2mn 2 7nm 2 If possible, collect like terms. If not possible, write “not possible”: 0.3xy 0.5 yx 7 2x x 9 5 3x If possible, collect like terms. If not possible, write “not possible”: 2 x 10 x 2 If possible, collect like terms. If not possible, write “not possible”: x 5 3 1 1 If possible, collect like terms. If not possible, write “not possible”: x x 2 2 If possible, collect like terms. If not possible, write “not possible”: If possible, collect like terms. If not possible, write “not possible”: 0.24 xy 0.17 x If possible, collect like terms. If not possible, write “not possible”: 1 3 x x 2 2 If possible, collect like terms. If not possible, write “not possible” : 0.5c 3 10c 4 If possible, collect like terms. If not possible, write “not possible”: 0.4 x 2 y 0.8 yx 2 18B Solving linear equations (inequalities) when the removal of parentheses is needed; no fractions involved. “No solution” or “all real numbers” as a solution is a possibility. Operations on decimals might be needed. 62 Solve the following equation: 3x 2 4( x 1) Solve the following equation: 6(2 x 3) x 11x Solve the following equation: ( x 3) x (2 x 1) Solve the following equation: 0.2( x 3) 0.2 x 3(0.1 x) Solve the following equation: 2(4 x 1) 7 x 4 x Solve the following equation: 4( y 2) (1 y ) Solve the following equation: 0 2(1 x) Solve the following equation: 4x 3 12 Solve the following equation: 6 2( x 5) Solve the following equation: 0.7 2( x 0.3) x 0.1 Solve the following equation: 32m 4 6m 6 Solve the following inequality: 6 2( x 5) Solve the following inequality: 3x 1 11 3( x 4) Solve the following inequality: 4 x 1 7 x (5 2 x) Solve the following inequality: x 1 6( x 2) Solve the following inequality: 4 x 1 2( x 5) Solve the following inequality: 3( x 2) 3x 2 Solve the following inequality: 3(2 x 5) 6 x 16 Solve the following inequality: 3 23a 5 6a Solve the following inequality: 32 x 1 2x 2 Solve the following inequality: 0.1(10 x 100) x 63 19B Solving equations for a given variable. Factorization of this variable might be needed in the process. Variable might be on both sides of the equation. Variable could be in the denominator. No simplification needed at the end. More than one operation might be needed in the process. Solve for x : x b ac Solve for x : 2x a b Solve for x : ax b c Solve for x : ax bx c 0 Solve for x : a b x Solve for the indicated variable: ax b 4 for a Solve for the indicated variable: m2 a for u . u Solve for the indicated variable: 2 x y t 3s for y . Solve for the indicated variable: AX A 1 for A. Solve for the indicated variable: x 3 sy 2 x for y. Solve for the indicated variable: 3v t s v for v . Solve for x : x a . Solve for x : ax by 3 x . 2m 4k n2 b Solve for b : c a Solve for m : Solve for b: abc 2 1 c Solve for x : m 2n a x 64 Solve for s: sx s t Solve for x : ax 2x 1 Solve the following equation x d e for y . y 20B Writing phrases as algebraic expressions (involving 2 operations). Use the letter x to represent a number and write the following as algebraic expression: Seven more than one third of a number Use the letter x to represent a number and write the following as algebraic expression: A quantity decreased by 9, and then multiplied by A Use the letter x to represent a number and write the following as algebraic expression: A number cubed, and then decreased by y Write a direct translation of the following phrases as an algebraic expression. Multiply 3 by x , and then add y. Write a direct translation of the following phrases as an algebraic expression. Multiply the sum of a and b by 4. Write a direct translation of the following phrases as an algebraic expression. Do not simplify. The opposite of x , then raise it to the sixth power. Write a direct translation of the following phrases as an algebraic expression. Do not simplify. Subtract 3 from y, and then multiply it by z . Write a direct translation of the following phrases as an algebraic expression. Raise x to the third power, and then multiply by 9. Write a direct translation of the following phrases as an algebraic expression. Do not simplify. Multiply x by 9, and then raise the result to the third power. Write a direct translation of the following phrases as an algebraic expression. The difference of a and b, then divided by c Write a direct translation of the following phrases as an algebraic expression. Divide 3 by y, and then add x Write a direct translation of the following phrases as an algebraic expression. 65 Raise x to the third power, raise y to the seventh power, and then add them together Let C be a variable representing the temperature in Celsius. Write the following phrase as an algebraic expression: Nine fifths of the Celsius temperature plus 32. Let m represents mass, and c speed of light. Use m and c to write the following phrase as an algebraic expression: the product of mass and the square of speed of light. Use the letter X to represent a number and write the following as an algebraic expression: A number decreased by 7, and then the result is doubled. Use the letter Y to represent a number and write the following as an algebraic expression: The opposite of a number, then raised to one hundred and twenty first power Use the letter C to represent a number and write the following as an algebraic expression: A number, first divided by 2, and then raised to the third power. Use the letter s to represent a number and write the following as an algebraic expression: Multiply a number by 9, and then subtract it from c. Use the letter m to represent a number and write the following as an algebraic expression: The difference between a number and 4, then multiplied by y. Use the letter X of your choice to represent a number and write the following as an algebraic expression: Take one fourth of a number, and then subtract 5. 21B Identifying linear equations and writing them in a standard form. Recognizing the value of parameters in the representation. Rewriting algebraic expressions to match a prescribed format and identifying the values of given variables. Determine if the following equation is linear in one variable. If it is not linear, write “not linear”. If it is linear, express it in the form ax b 0 , a 0 . Determine the values of a and b in your x representation: 1 0 4 Determine if the following equation is linear in one variable. If so, express it in the form ax b 0 , a 0 . Determine the values of a and b in your representation: 7 x 2 3x 66 Determine if the following equation is linear in one variable. If it is not linear, write “not linear”. If it is linear, express it in the form ax b 0 , a 0 . Determine the values of a and b in your representation: 2 x( x 3) 2 x 2 Determine if the following equation is linear in one variable. If it is not linear, write “not linear. If it is linear, express it in the form ax b 0 , a 0 . Determine the values of a and b in your representation: 3x 2 1 0 Write the expression ( x 2) 2 ( y 1) 2 r 2 in the form: ( x p) 2 ( y q) 2 r 2 are the values of p and q? What Write the following expression in the form A 2 , where A is any algebraic expression. Identify 81x 2 A: Write the following expression in the form A 2 , where A is any algebraic expression. Identify x8 A: Write the following expression in the form A 2 , where A is any algebraic expression. Identify 0.09 x 2 A: y6 Write the following expression in the form Ax By , where A , B are any numbers. Identify A x 8y and B in your representation: 4 Write the following expression in the form A5 , where A is any algebraic expression. Identify A: x 10 Write the following expression in the form A 2 , where A is any algebraic expression. Identify A: 36 x 2 y 2 Write the following equation in the form y mx b , where m and b are any numbers. In the y 2x 4 expression you obtained, identify m and b : Write the following equation in the form y mx b , where m and b are any numbers. In the yx0 expression you obtained, identify m and b : Write the following equation in the form y mx b , where m and b are any numbers. In the 2 y 4 x 3 expression you obtained, identify m and b : Write the expression x 24 in the form a 4 . Identify a in your representation. Write the expression x 24 in the form a 6 . Identify a in your representation. 67 Rewrite the following expression in the form ax 3 b , where a and b are any numbers. Identify a and b in your representation: x 3 3 Rewrite the following expression in the form ax 3 b , where a and b are any numbers. Identify a and b in your representation: 2 x 3 3 Rewrite the following expression in the form ax 3 b , where a and b are any numbers. Identify 4x3 3 a and b in your representation: 2 Rewrite the following expression in the form ax 3 b , where a and b are any numbers. Identify (2 x) 3 4 a and b in your representation: Write the following expression in the form a 3 , where a is any algebraic expression or a number. State what a is equal to: 27 Write the following expression in the form a 3 , where a is any algebraic expression or a number. 8 State what a is equal to: 125 Write the following expression in the form a 3 , where a is any algebraic expression or a number. State what a is equal to: 64x 3 Write the following expression in the form a 3 , where a is any algebraic expression or a number. State what a is equal to: y6 22B Applying laws of exponents in order to evaluate an expression (up to two operations). Performing an operation on fractions or on decimals might be needed. Or Factoring -1 Evaluate: 9 81 9 80 Evaluate: 47 46 411 68 15 248 15 247 (12 3 ) 6 Evaluate: 1217 3 21 310 Evaluate: 3 29 10 0.5 0.5 6 Evaluate: (0.5 7 ) 2 Evaluate: (64 3 )11 Evaluate: 64 32 215 216 Evaluate: 27 2 Factor 1 from the following expression: 2 x y z . X Factor 1 from the following expression: y. 3 Factor 1 from the following expression: a b 1 x yz Factor 1 from the following expression: a 2 Factor 1 from the following expression: 4 x 3 5 y 2 5 z 1 1 s Factor 1 from the following expression: ab ab 2 2 4 Factor 1 from the following expression: x y z 3 3 Factor 1 from the following expression: 2( x 2 y) ( x 2 z ) z 2 23B Removing parentheses from expressions of the form (a b) 2 and simplifying them by collecting like terms. Eliminate parentheses and simplify by collecting like terms: (2 3x) 2 Eliminate parentheses and simplify by collecting like terms: (3ab 1) 2 Eliminate parentheses and simplify by collecting like terms: (2 x y ) 2 Eliminate parentheses and simplify by collecting like terms: ( xy 3) 2 69 Eliminate parentheses and simplify by collecting like terms: (3a b) 2 Eliminate parentheses and simplify by collecting like terms: (2 x 3 ) 2 Eliminate parentheses and simplify by collecting like terms: (0.1 x) 2 Eliminate parentheses and simplify by collecting like terms: (a 2 b 1) 2 Eliminate parentheses and simplify by collecting like terms: (7 2m) 2 Eliminate parentheses and simplify by collecting like terms: ( xyz 1) 2 Eliminate parentheses and simplify by collecting like terms: (0.5 y 5 ) 2 Eliminate parentheses and simplify by collecting like terms: (0.2 x 0.1y) 2 Eliminate parentheses and simplify by collecting like terms: (8 x 2 y) 2 Eliminate parentheses and simplify by collecting like terms: (3m n) 2 Eliminate parentheses and simplify by collecting like terms: (2s 3t ) 2 24B Evaluation of an algebraic expression when the value of a part of the algebraic expression is given (rather than the value of the variables). No rewriting the expression in its equivalent form is needed. If A 5 3 , evaluate: 2A5 If A 5 3 , evaluate: (2 A5 ) 2 If A 5 3 , evaluate: 2 2 A5 2 a a 3 Evaluate the expression , if b b a a 3 Evaluate the expression , if b b ab ab 2 , if 2 Evaluate the expression c c 70 ab Evaluate the expression c 2 ab Evaluate the expression c 2 , if ab 2 c , if ab 2 c If x y 2 , evaluate ( x y) 2 4 If x y 2 , evaluate If x y 1 7 3 1 2 1 , evaluate A 7 A Evaluate the expression 1 , if GHJ 1 : GHJ Evaluate the expression 2 Evaluate the expression a a 10 , if 3 b b xy , if z xy 1 z 3 Evaluate the following expression a b c d if a b 2 cd 3 Evaluate the following expression 9(a b) 2 c d if a b 2 Evaluate the following expression ab if a b 9 cd cd 3 cd 3 Evaluate the expression ( x z ) x 2 y if x 2 y 0.2, x z 0.6 Evaluate the expression xz if x 2 y 0.2, x z 0.6 2 x y Evaluate the expression ( x z ) x 2 y if x 2 y 0.2, x z 0.6 Evaluate the expression 4 xyzt , if xy 1, zt 3 . Evaluate the expression xy zt , if xy 1, zt 3 . 71 Evaluate the expression xy zt , if xy 1, zt 3 . xy Evaluate the following expression y 4 x 3 z 6 , if x 3 y 4 z 6 Evaluate the following expression 2 . 3 1 , if x 2 y 2 0.1 x y2 2 Evaluate the following expression y 2 x 2 0.3 , if y 2 x 2 0.1 DIFFICULT CATEGORY: randomly select 4 categories (1 question from each category) 1D Ability to rewrite the expression by substituting (up to 2 substitutions) value of variable(s) and then evaluating the expression, if possible. If evaluation is not possible, recognizing that it is not possible to evaluate. “More challenging” examples. The expression all of them. a) x 3, 3 x cannot be evaluated for which of the following values of x and y? Circle y 5 y 5 The expression 3x y5 c) x 3, y0 d) x 0, y5 1 cannot be evaluated for which of the following values of x and y? 2 y Circle all of them. a) x 3, y 2 The expression b) x 3, b) x 3, y 2 c) x 3, y 0 d) x 0, y 2 3y cannot be evaluated for which of the following values of x and y? Circle 2 x all of them. a) x 0, y2 The expression 3x cannot be evaluated for which of the following values of x and y? Circle yx all of them. a) x 3, y3 b) x 3, b) x 5, y0 y5 c) x 2, c) x 0, y2 d) x 0, y 5 d) x 2, y0 y0 Rewrite the expression by substituting values of variables and then evaluate it, if possible. If not possible, write “undefined”: ( A 3B ) , if A 1, B 1 72 Rewrite the expression by substituting values of variables and then evaluate it, if possible. If not possible, write “undefined”: ( A 3B ) , if A 0.1, B 0.2 Rewrite the expression by substituting values of variables and then evaluate it, if possible. If not 2 1 possible, write “undefined”: ( A 3B ) , if A , B 7 6 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not possible, write “undefined”: 2a 2 (2a) 2 if a 1 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not 1 2 possible, write “undefined”: A 2 B 2 when A , B 3 3 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not A B 1 2 possible, write “undefined”: when A , B A B 3 3 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not A( B) 1 2 possible, write “undefined”: when A , B AB 3 3 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not 0.1x possible, write “undefined”: 22 when x 2, y 0.02 y Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not x y possible, write “undefined”: when x 0.1, y 0.2 y x Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not possible, write “undefined”: m (n 4) 2 when m 1, n 2, Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not A B 5 A 4, B possible, write “undefined”: 3 9 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not A 2 possible, write “undefined”: AB B A , B 1 B 8 73 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not 1 possible, write “undefined”: (ab) 2 a b a 1, b 3 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not bn 1 possible, write “undefined”: b ,n3 bn 1 3 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not possible, write “undefined”: n 0.2 m when m 0.3, n 0.6 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not 1 4 possible, write “undefined”: 8m 10n when m , n 8 5 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not 1 1 4 possible, write “undefined”: n ( m) when m , n 8 8 5 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not 1 4 possible, write “undefined”: 8m 2 n when m , n 8 5 Rewrite the expression by substituting value of a variable and then evaluate it, if possible. If not possible, write “undefined”: 2(m 3n) when m 2, n 5 2D Recognizing equivalent expressions. Applying more than one of the following rules at a time. a) the order of terms of an expression can be changed b) the order of factors of an expression can be changed b c bc a a a a a a d) b b b ab b 1 a ab e) c c c c) Circle all of the following expressions that are equivalent to m (n k ) : (n k ) ( k n) m , , , m mnk , mnk , 1 74 Circle all of the following expressions that are equivalent to a 4c , b 1 (4c a) , b 1 (1)( a 4c), b a 4c : b 4c a , b b Rewrite the following expression in its equivalent form as a single fraction: x a a2 2 4s 4s Rewrite the following expression in its equivalent form as a single fraction: 7 Rewrite the following expression in its equivalent form as a single fraction: y 1 m 3 1 nm 2 k t k t Rewrite the following expression in its equivalent form as a single fraction: x y zt 2 ( a b) t t Rewrite the following expression in its equivalent form as a single fraction: 2a mn 3 xy xy Rewrite the following expression in its equivalent form as a single fraction: 3 a b cd 1 2 t t Circle all of the following expressions that are equivalent to xy ab(c d ) : ab(d c) xy (c d )ab yx ab(c d )( yx) Circle all of the following expressions that are equivalent to b ad , 6 1 (b ad ), 6 da b, 6 Circle all of the following expressions that are equivalent to ad b , 6 6 b da , 6 ad b : 6 b da 6 (ad b) 6 m n : x x ( m n) x Circle all of the following expressions that are equivalent to m n , x x , 1 (m n) , x 75 Circle all of the following expressions that are equivalent to a2 , 3b (2 a) , 3b a2 3b 1 a b, cd , a2 3b 1 (a 2)( 1) 3b Circle all of the following expressions that are equivalent to 2 a2 3b a 2 , (3 b) (3 b) Circle all of the following expressions that are equivalent to (2 a) , (3 b) 2( a b ) cd (b a) 2, (c d ) 2(b a) (d c) , cd , 2( a b ) Replace with expression such that the resulting statement is true. Replace with expression such that the resulting statement is true: 2x 7 y 7y ab ab mn ( m n) v Replace with expression such that the resulting statement is true: 2(a 2b)(3 d ) (3 d ) Replace with expression such that the resulting statement is true: da b b d m mn 2 2 a a Replace with expression such that the resulting statement is true: b Replace with expression such that the resulting statement is true: 3D Simplifying expressions involving exponential expressions (“relatively difficult examples”). Operations on fractions or decimals might be needed. 20 a Circle all expressions that are equivalent to : 3 20 20 12 8 a a (a ) a 12 a 8 , , , , 3 3 20 3 20 3 20 (a 4 ) 5 3 20 76 Perform the indicated operations and simplify: y5 3 y y8 Perform the indicated operations and simplify: (2a 2 b)( a 3 ) Perform the indicated operations and simplify: 2a Perform the indicated operations and simplify: a7 (2a) 2 2x3 x5 x4 (3x 2 ) 3 Perform the indicated operations and simplify: x4 Perform the indicated operations and simplify: a4 2 a a4 2 1 Perform the indicated operations and simplify: v 7 v Perform the indicated operations and simplify: (3aa 2 ) 2 x2 y Perform the indicated operations and simplify: x 3 Perform the indicated operations and simplify: 2 x(2 x 5 y 3 ) 3 Perform the indicated operations and simplify: 3b(10b 5 ) 2 Perform the indicated operations and simplify: 3a(ab 2 ) 8 a3 Perform the indicated operations and simplify: 2 4b 2 Perform the indicated operations and simplify: x 2 y( x 3 ) y 4 Perform the indicated operations and simplify: s2x (4s) 2 Perform the indicated operations and simplify: ab 7 a ba 2 b 6 77 Perform the indicated operations and simplify: 2x 2 y 2 y x Perform the indicated operations and simplify: a13b 2 c 4 a 5c 2b Perform the indicated operations and simplify: (ab) 8 b 2ab 4 4D Applying the Distributive Law when removing parentheses in multiplication of a binomial by a trinomial. Collecting terms is needed. Eliminate parentheses and simplify: (b c 2)(b 2 b 3 ) Eliminate parentheses and simplify: ( x 3 2 x 2 2 x)( x 2 x 3 ) Eliminate parentheses and simplify: ( x 2 y 2 x y)( x 2 3) Eliminate parentheses and simplify: (a 0.2)( a 0.1b ab) Eliminate parentheses and simplify: ( x 3 2 x 2 2 x)( x 2 x 3 ) Eliminate parentheses and simplify: ( x 3 y 2 2 x 2 3)( x 3 y 2 1) Eliminate parentheses and simplify: (4m m 2 )(2 m 3m 2 ) Eliminate parentheses and simplify: (2 A 3 4 B)(5 A B) Eliminate parentheses and simplify: (m 3 5m 2 2m)(3m 2 m 3 ) Eliminate parentheses and simplify: (ab 3 ab 2 ab)(ab 2 ab 3 ) Eliminate parentheses and simplify: (u 9t )(ut u 2 5t 2 ) Eliminate parentheses and simplify: ( x 3 y 2 x 2 3)(1 y 3 ) Eliminate parentheses and simplify: (a b)( a b 3) Eliminate parentheses and simplify: ( x y )( x y 2) 78 Eliminate parentheses and simplify: (a 2 b 2 0.8)(b 2 a 2 ) Eliminate parentheses and simplify: (m 3 n 2 2m 2 n 3 1)(2 m 2 n 3 ) Eliminate parentheses and simplify: (2s 2 t )(st st 2 3ts 3 ) Eliminate parentheses and simplify: (u 3 3u 2 4u)(2u 2 10u 3 ) Eliminate parentheses and simplify: (2m 3 n 2 3)(m 3 n 2 1 m 6 n 4 ) Eliminate parentheses and simplify: ( x 7 y 2 x 2 2 yx 5 )( x 2 x 5 y) 5D Factoring an indicated expression involving parentheses, fractions or decimals. Factor (a 2b) from the following expression: 4 x(a 2b) (a 2b) . 2 from the following expression: 3 Factor Factor 0.01 Factor Factor 2 2 4 x y z 3 3 from the following expression: 0.2 x 2 0.03 z 1 1 1 s from the following expression: ab ab ab 1 from the following expression: 5 1 1 x 5 25 Factor a b from the following expression: 6(a b) x(a b) Factor a b from the following expression: 4(a b) 3(a b) 2 Factor x 2x x from the following expression: a y y y Factor 0.02 from the 0.2lw 10lh 0.04wh Factor 11 11 2 33 t t from the following expression 2 2 2 79 Factor m 3n from the following expression: 5x(m 3n) 4(m 3n) 5 Factor s t 2 from the following expression s(s t 2 ) t ( s t 2 ) Factor 1.6 from the following expression 1.6 x 3.2 x 2 16 x 3 Factor 0.001 from the following expression 0.1a 4 0.01a 3 0.001a 4 Factor 3 from the following expression: 8 1 2 5 Factor 6 2 Factor 3 9 Factor 5 Factor from the following expression: from the following expression: from the following expression: from the following expression: 6 9 s t2 8 8 5 2 1 x yz 2 2 5 5 5 ab bc ac 6 6 6 4 5 z 3 3 9 27 2 y y 5 10 6D Rewriting an algebraic expression in terms of a designated variable when a rewriting of the expression in an equivalent form is needed. Evaluation of an algebraic expression when the value of a part of the algebraic expression is given (rather than the value of the variables) and rewriting the expression in its equivalent form is needed. Rewrite the expression a 2b 3c 4d in terms of x , if it is given that a 3c 5x and 4d 2b x . Simplify. Rewrite the expression 7 x 7 y in terms of m if x y m . Rewrite the expression x y in terms of m if x y m . Rewrite the expression x y in terms of m if x y m . 7 7 Rewrite the expression 1 1 x. in terms of x if 2 A A 80 Rewrite the expression c b 2a in terms of x if 2a b c x . Rewrite the expression 2a b c in terms of x if 2a b c x . Rewrite the expression a b c a in terms of x if 2a b c x . Rewrite the expression m(2)n(1) p in terms of s if mnp s . Simplify. Rewrite the expression m 3 (np) 3 in terms of s if mnp s . Rewrite the expression a2a a y . in terms of y if 3 b b x2 y2 xy m in terms of m if . 2 z 2 z Rewrite the expression Rewrite the expression zxty in terms of m if xy m, zt 3m . Simplify. Rewrite the expression z 6 (2) y 4 (3) x 3 in terms of s if x 3 y 4 z 6 s . Simplify. Rewrite the expression 1 2 1 2 x y in terms of s if x 2 y 2 2s . Simplify. 2 2 Write the expression 6 a Write the expression b in terms of m, if ab m a b in terms of m, if ab m 2 1 Write the expression (2b)( a ) in terms of m, if ab m 2 Write the expression a 3b 3 in terms of m, if ab m Write the expression 2z 1 x t x t in terms of y, if y , 2 y . Simplify. z z z z Write the expression 2x t x t in terms of y, if y , 2 y . Simplify. z z z z 81 Write the expression xt x t in terms of y, if y , 2 y . Simplify. z z z 7D Solving linear equations (inequalities) involving fractions. “No solution” or “all real numbers” as a solution is a possibility. Solve the following equation: Solve the following inequality: Solve the following inequality: Solve the following inequality: Solve the following inequality: Solve the following inequality: Solve the following inequality: Solve the following inequality: Solve the following inequality: Solve the following inequality: Solve the following inequality: Solve the following equation: Solve the following equation: Solve the following equation: Solve the following equation: 4x 2 1 5 3 4 x x2 5 2 4 3x 6 x2 3 x2 5 4 y y3 1 2 3 2a 3 2 1 5a 5 3 x 1 1 x 4 x 1 x 1 4 3 3x x 15 4 8 x 5 2x 4 3 x x 2 5 2 3 x2 5 x 4 x 3x 1 0 5 x 1 x 1 3 2 x 1 8x 2 82 1 1 3 y y 5 5 10 x 2 3 Solve the following equation: 2 3 4 3x 3x 1 Solve the following equation: 2 5 2 5 Solve the following equation: 1 a 3 6 Solve the following equation: 8D Solving equations for a given variable. Factorization of this variable might be needed in the process. Variable could be in the denominator. Simplification at the end required. Solve for the indicated variable. Simplify your answer whenever possible: b ac for b a Solve for the indicated variable. Simplify your answer whenever possible: ax b 4b for a Solve for the indicated variable. Simplify your answer whenever possible: abc 2 (ac) 3 for b Solve for the indicated variable. Simplify your answer whenever possible: AX A X 1 for A a2 a for u Solve for the indicated variable. Simplify your answer whenever possible: u Solve for the indicated variable. Simplify your answer whenever possible: xa a 2 3a 2 for x Solve for the indicated variable. Simplify your answer whenever possible: ax 3ax 4 for x Solve for the indicated variable. Simplify your answer whenever possible: s 4t for s. t3 Solve for the indicated variable. Simplify your answer whenever possible: 2t 8t 9t 2 tx for t. Solve for the indicated variable. Simplify your answer whenever possible: 2 ys xys y for s. Solve for the indicated variable. Simplify your answer whenever possible: Solve for the indicated variable. Simplify your answer whenever possible: mx m 2 n 3 for x. n a b for a b3 c 83 Solve for the indicated variable. Simplify your answer whenever possible: a2 a for u u Solve for the indicated variable. Simplify your answer whenever possible: 2m 4n for m n2 Solve for the indicated variable. Simplify your answer whenever possible: 2 x y t 3x for y ‘ Solve for the indicated variable. Simplify your answer whenever possible: x 3 xy 2x 3 for y Solve for the indicated variable. Simplify your answer whenever possible: s( x 1) s 2 for x Solve for the indicated variable. Simplify your answer whenever possible: ax by 3ax 4by for a Solve for the indicated variable. Simplify your answer whenever possible: 3(v s ) s for v Solve for the indicated variable. Simplify your answer whenever possible: ax x(a 2) 1 for x 9D Evaluating algebraic expressions when the value of a part of it is given (rather than the value of variables). Rewriting the original expression in some equivalent form is required to make the needed substitution. Evaluate the expression a c b , if a b 1 and c 2 . Evaluate the expression a b , if a b 1 and c 2 . c c Evaluate the expression 2c (b a) , if a b 1 and c 2 . Evaluate the expression m2 m 4. , if 2 n n Evaluate the expression n m 4. , if m n If x y 2 , evaluate 7 x 7 y 84 If x y 2 , evaluate x y 7 7 If x y 2 , evaluate x y . Evaluate the expression x 3 ( y 2 z 3 ) 2 , if x 3 y 4 z 6 2 . 3 Evaluate the expression z 6 (2) y 4 (3) x 3 , if x 3 y 4 z 6 Evaluate the expression 2 . 3 x3 y 4 6 2 z , if x 3 y 4 z 6 . 3 3 Evaluate the expression 2(a d c b) if a b 2 Evaluate the expression 9(a b) 2 d c if a b 2 cd 3 cd 3 Evaluate the expression, zxty if xy 1, zt 3 : Evaluate the expression 1 2 1 2 x y , if x 2 y 2 0.1 2 2 Evaluate the expression x2 y2 , if x 2 y 2 0.1 0.2 0.2 Evaluate the expression G 3 (HJ ) 3 , if GHJ 1 Evaluate the expression G (2) H (1) J , if GHJ 1 If 1 2 1 , evaluate 2 A 7 A If 1 2 1 , evaluate A 7 A Simplify the expression 4 x ( z 4 x) , and then evaluate when z 3 : VERY DIFFICULT CATEGORY: randomly select 1 category (1 question from the category selected) 85 1VD Writing phrases as algebraic expressions and then simplifying them: a) using the laws of exponents b) removing parentheses and collecting like terms. Write as a single algebraic expression using parentheses where appropriate Then eliminate the parentheses and simplify: The product of x and x 2 , then raised to the sixth power. Write as a single algebraic expression using parentheses where appropriate. Then eliminate the parentheses and simplify: The quotient of 4b 3 and b 2 , then raised to the second power Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and collect like terms: Add 3x 2 and 4x 2 and then raise the result to the second power/ Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and collect like terms: Subtract x 2 3 x 2 from 4 x 2 4 x 2 . Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and collect like terms: Multiply 2 x 2 3 x 2 and 4x 2 Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and 1 2 collect like terms: The product of a and 2a 3 5 Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and collect like terms: The product of 4, 2 x 2 y and 3 y x 2 Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and collect like terms: The sum of mnk , 4nmk , and 3mn Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and collect like terms: The sum of 3x and 2, then raised to the second power Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and collect like terms: The difference of 2a and b , then raised to the second power Write the following expression using algebraic symbols, eliminate parentheses, if necessary, and collect like terms: The product of 2 and 4x 1 , then added to 5x 2 Write as an algebraic expression using parentheses where appropriate, then remove the parentheses and simplify: The product of 3x and x 2 , then raised to the third power 86 Write as an algebraic expression using parentheses where appropriate, then remove the parentheses and simplify: The quotient of 4a 12 and a 2 , then raised to the second power Write as an algebraic expression using parentheses where appropriate, then remove the parentheses and simplify: a 3 raised to seventh power, then multiplied by a Write as an algebraic expression using parentheses where appropriate, then remove the parentheses and simplify: xy 7 raised to the fifth power , then divided by y 3 Write as an algebraic expression using parentheses where appropriate, then remove the parentheses and simplify: 3ab 3 squared, and then multiplied by b . Write as an algebraic expression, rewrite without parentheses and collect like terms, if possible; 2 y Subtract xy from 3 yx y 3 5 Write as an algebraic expression, rewrite without parentheses and collect like terms, if possible; The sum of 3 x 4 x 2 1 and twice the expression 4 x 4 2 2VD Rewriting algebraic expressions in terms of a designated variable requiring “challenging” simplifications. Express (m n)( m 2 mn n 2 ) in terms of x , if m 2 , and n 3x . Simplify. Express (m n)( m 2 mn n 2 ) in terms of m , if n m . Simplify. Write the expression (mn) 2 mn 2 in terms of m, if n m . Simplify. Write the expression (mn) 2 mn 2 in terms of s , if m 2s, n s . Simplify. Write the expression m 3 n 2m 2 n 2 in terms of s , if m s, n 2s . Simplify. Write the expression m 3 n 2m 2 n 2 in terms of s , if m 2s 2 , n s 4 . Simplify. Write the expression m 3 n 2m 2 n 2 in terms of s , if n m s . Simplify. Write the expression (m n) 2 (m n) 2 in terms of s , if m 2s, n 3s . Simplify. Write the expression (m n) 2 (m n) 2 in terms of m, if n m . Simplify. 87 Write the expression (m n) 2 (m n) 2 in terms of m, if n m . Simplify. Rewrite the expression (a b)( a b) in terms of x , if it is given that a 5x and b 2 x . Simplify. 3VD “Non standard” factoring of the difference of squares, difference (sum) of cubes (formula for the difference/sum of cubes given but students need to know the formula for the difference of squares). Factor and then simplify the following expression m 2 (1 3m) 2 Factor and then simplify the following expression m 2 (2m 1) 2 Factor and then simplify the following expression ( x 1) 2 (3x 5) 2 Factor and then simplify the following expression (2a 3) 2 9a 2 Factor and then simplify the following expression (a 2b 3) 2 9 Factor and then simplify the following expression ( x 2 3 y 2 ) 2 (2 x 2 1) 2 Factor and then simplify the following expression (3x 2) 2 ( y 5x 1) 2 a 3 b 3 (a b)( a 2 ab b 2 ) . Factor the following The following formula is true: expression using the above formula: y 3 x 3 64 . a 3 b 3 (a b)( a 2 ab b 2 ) . Factor the following 1 8y3 expression using the above formula: 3 a The following formula is true: a 3 b 3 (a b)( a 2 ab b 2 ) . Factor the following The following formula is true: expression using the above formula: y 6 1 . a 3 b 3 (a b)( a 2 ab b 2 ) . Factor the following The following formula is true: 27 x 3 8 y 3 expression using the above formula: 88 The following formula is true: a 3 b 3 (a b)(a 2 ab b 2 ) . Factor the following expression using the above formula: y3 125 x 3 8 . The following formula is true: a 3 b 3 (a b)( a 2 ab b 2 ) . Factor the following expression using the above formula: 64 ( xy) 3 a 3 b 3 (a b)(a 2 ab b 2 ) . Factor the following 1 27 y 3 expression using the above formula: 8 The following formula is true: The following formula is true: a 3 b 3 (a b)(a 2 ab b 2 ) . Factor the following expression using the above formula: 8 (mn) 3 4VD Replacing up to 2 variables with an algebraic expression and solving the resulting equation. Let a 3x 2 and b x 2 . Find the value of x so that the following is true: ab Let a 3x 2 and b x 2 . Find the value of x so that the following is true: a 2b Let A 2 x, B x, . Find x , so the following are true: A B 1 3 8 Let A 2 x 1, B x 2, . Find x , so the following are true: A B 0 Let A 2 x, B x, . Find x , so the following are true: A 3B 1 Let P 3( x 1), Q 4 x 5 . Find x such that the following is true: P Q Let P 3( x 1), Q 4 x 5 . Find x such that the following is true: Q P 2 Let x 3a 2, y 2a 1 . Find a , so the following are true: x y 3 y z Let y 2a 1, z 2 a . Find a , so the following are true: 2 4 89

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