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Variable Expressions Vocabulary Words To Know Translating Words to Variable Expressions 1. The SUM of a number and nine 2. The DIFFERENCE of a number and nine n+9 n–9 3. The PRODUCT of a number and nine 4. The QUOTIENT of a number and nine 9n 5. One ninTH OF a number 6. Nine times THE QUANTITY OF a number increased by ten. 9(n + 10) or 7. A number SQUARED n2 9. 8. A number CUBED n3 A number LESS THAN nine n – 9 * When you see the phrase less than, reverse the terms. Translating Variable Expressions Translate each mathematical expression into a verbal phrase without using the words: “plus”, “add”, “minus”, “subtracted”, “”take–away”, multiplied”, “times”, “over”, or “divided” . 16. 13a 17. 14 a the product of thirteen and a number the quotient of fourteen and a number eleven less than a number , OR a number decreased by eleven, OR the difference of a number and eleven 18. y – 11 19. 3y + 8 the product of three and a number increased by eight, OR 20. 6 ÷ n2 21. 7(x + 1) eight more than the product of a number and three 3 22. b – 4 the quotient of six and a number squared seven, times the quantity of a number increased by one a number cubed decreased by four or four less than a number cubed or the difference of a number cubed and four Simplifying Using Order of Operations 1. 2 (1 + 3) • 4 6 – 2 ÷ (–1) 2 ( 4 ) •4 16 •4 64 6 – 2 ÷ (–1) 6 – (–2) 6 + 2 8 64 = 8 8 Evaluate the 2. Evaluate inside numerator and –4 + 8 – (5 + 9) •2 the brackets denominator first... –4 + 8 – ( 14 ) •2 separately [ [ –4 +[ –6 –4 + ] ] ]•2 –12 –16 ...then treat the brackets like parenthesis Evaluating Variable Expressions with Negative Variables 1) –x Evaluate each expression using: –(–3) +3 3 2) –y –(–2) +2 2 3) x = –3 y = –2 z=6 1. Substitute –3 for x only. 2. Leave the negative (–) in front of the x alone. 3. Now, simplify the signs (kill the sleeping man) 1. Substitute –2 for y only. 2. Leave the negative (–) in front of the y alone. 3. Now, simplify the signs. –z –6 –6 1. Substitute 6 for z only. 2. Leave the negative (–) in front of the z alone. 3. Now, simplify the signs. Evaluating Variable Expressions with Negative Variables 4) x–y –3 – (–2) –3 + 2 –1 5) x = –3 1. 2. 3. 4. y = –2 z=6 Substitute –3 for x , and –2 for y only. Leave the subtraction sign (–) in front of the y alone. Now, simplify the signs. (keep->change->change) Add the integers. x–z –3 – 6 –9 6) Evaluate each expression using: 1. Substitute –3 for x , and 6 for z only. 2. Leave the subtraction sign (–) in front of the z alone. 3. Subtract the integers. z–x 6 – (–3) 6 +3 9 1. 2. 3. 4. Substitute 6 for z , and –3 for x only. Leave the subtraction sign (–) in front of the x alone. Now, simplify the signs. Add the integers. Evaluating Variable Expressions with Negative Variables xy 7) –3 • (–2) 6 Evaluate each expression using: 1. 2. x = –3 y = –2 z=6 Substitute –3 for x , and –2 for y. Multiply –– * Why? Two variables right next to each other. yz 8) –2 • 6 –12 1. 2. Substitute –2 for y , and 6 for z. Multiply –– * Why? Two variables right next to each other. –xz 9) –(–3) • 6 +3 • 6 18 10) 1. 2. 3. 4. Substitute –3 for x , and 6 for z. Leave the negative sign in front of the x alone. Simplify the signs. Multiply 1. 2. 3. 4. Substitute –3 for x , and 6 for z. Leave the negative sign in front of the parenthesis, ( ), alone. Multiply inside the parenthesis first. Simplify the signs. –(xz) –((–3) • 6) –(–18) 18 Evaluating Variable Expressions with Negative Variables 11) 2x2 2 • (–3)2 2• 9 18 12) x = –3 36 z=6 Substitute –3 for x. First, evaluate the exponent. Then, multiply. –– Why? When a number is right next to a variable, multiply. 1. 2. 3. Substitute –3 for x. First, evaluate the exponent. Then, multiply. 1. 2. 3. Substitute –3 for x. First, evaluate inside parenthesis, ( ). Then, evaluate the exponent. (–2x)2 (–2 • (–3))2 ( 6 )2 y = –2 1. 2. 3. –2x2 –2 • (–3)2 –2 • 9 –18 13) Evaluate each expression using: Evaluating Variable Expressions with Negative Variables Evaluate each expression using: 14) 1. 2. 3. –2x3 –2 • (–3)3 –2 • (–27) 54 16) y = –2 z=6 2 x3 2 • (–3)3 2 • (–27) –54 15) x = –3 Substitute –3 for x. First, evaluate the exponent. * Remember, (–3)3 is (–3)•(–3)•(–3) = –27 Then, multiply. –– Why? When a number is right next to a variable, multiply. 1. 2. 3. Substitute –3 for x. First, evaluate the exponent. Then, multiply. 1. 2. 3. Substitute –3 for x. First, evaluate inside parenthesis, ( ). Then, evaluate the exponent. (–2x)3 (–2 • (–3))3 ( 6 )3 216 Evaluating Variable Expressions 1. 2. 4 7 3. –22 4. 5. 0 11 6. 21 7. 40 59 Simplifying Variable Expressions by Adding or Subtracting You can only add or subtract –7a – 9 + 11a –3 4a – 12 Circle the variable terms, ... ... and box up the constants Add the like terms. 1. 4. 17a + a Remember, a = 1a so, put a “1” in front of the a 2. 14x + 7b – 9x + 19 – 11b – 21 5. –10 –7 y + 6 y – 3 –13 –1y 13 + 2(8 – g) 13 + 16 + 2g – 4b + 5x – 2 29 + 2g 3. ... or, get rid of the “1” Use Distributive Property to get rid of the parenthesis. outer times first, then outer times second 12b + 5 – 15 – 12b 0 –10 6. ... or, get rid of the “0” 13 +(– 19) – 6(n + 1) – 10n 13 +(– 19) – 6n – 6 – 10n –12 – 16n Simplifying Variable Expressions by Multiplication 8. 7( –3x ) 7( –3x ) When you see constants (7 and –3) and variables (x), it’s easiest to simplify them separately. First, multiply 7 and –3… …then just bring down the x (Why? It’s the only x ) –21 x 9. 10. –19a • 10bc –19a • 10bc When you see constants (–19 and 10) and multiple variables (a, b, and c), take it one at a time. –190 abc First, multiply –19 and 10… …then bring down the a, b, and c (Why? There’s only 1 of each.) ( –1 )2y (–1)2y –2y 11. –5a( –5c ) –5a(–5c) 25ac 12. ( x • 8 )6y (x • 8)6y 48xy Simplifying Variable Expressions Using the Distributive Property 13. When a number or variable term sits right next to terms inside parenthesis, use the Distributive Property to simplify. –2(n +1) –2 •n –2n –2 • +1 –2 How? First, multiply the outer term, –2, by the 1st term in parenthesis, n. Then, multiply the outer term, –2, by the 2nd term in parenthesis, 1. 14. (9 – 6x)3 15. –4(8a + 7) 27 – 18x –32a – 28 How to remember the Distributive Property? 16. (–3p + 1)(–5) 15p – 5 17. 10(–c – 6) –10c – 60 “Outer times 1st, then outer times 2nd” Simplifying Variable Expressions by 18. Simplify x4 • x7 Multiplying Exponents Are the bases, x, the same? Are we multiplying or dividing the exponent terms? x 4 + 7 = 11 x 19. Multiplying Exponents Rule: When multiplying exponent terms with like bases, keep the base, then add the exponents. a6 • a9 a 15 So, we’re going to keep the base... …then add the exponents 11 Rewrite. 20. 1 b • b5 b6 Careful: What’s the invisible exponent over b ? 21. y • y4 • y4 9 y Simplifying Variable Expressions by GUIDED PRACTICE Multiplying Exponents Simplify 22. 2 4 7x • 7x 7 x2 • 7 x4 When you see both constants, 7, and variables, x, it’s easiest to simplify them separately. Let’s multiply the 7’s first... 49 x 23. 6 10y7 • 4y 7 10y • 4y 40y8 … then, multiply x2 • x4 . 24. 3a5b • 3a6b8 Don’t panic: Just multiply each part separately. 3a5b • 3a6b8 9a11b9 25. 2x5yz3 • yz 2x5yz3 • yz 2x5y2z4 Simplifying Variable Expressions by Dividing Exponents Simplify Are the bases, x, the same? 12 26. x 7 x Are we multiplying or dividing the exponent terms? Dividing Exponents Rule: When dividing exponent terms with like bases, keep the base, then subtract the exponents. x 12 – 7 = 5 x 5 y9 27. y3 6 y So, we’re going to keep the base... …then subtract the exponents Rewrite. 28. 4 a ÷a a 3 9 b 29. b8 b n (huh?) 30. 7 n 1 –6 n or n 6 Simplifying Variable Expressions by Dividing Exponents 31. Simplify When you see both constants, 12 and 6, and variables, y, it’s easiest to simplify them separately. 12 y 9 6 y3 12 y 9 6 y3 Let’s simplify the fraction … then, divide y9 and y3 . 2 y6 32. 12first... 6 45a 2 33. 9a 8 32 n 10 n 6 34. 5 p12 15 p 8 35. 28 xy5 z 2 36. 4 xy7 23d 2 d 7f3 Hint: The rest of the answers are fractions. 5a 16n 5 2 4 p 3 7z2 y2 23d 3 7f3