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Algebra 1 Chapter 7 Notes Chapter 7: Exponents Name: ___________________ Section 7.1: Negative and Zero Exponents A. Review of Exponents baseexponent The _______________ represents the number of times the __________ is multiplied by itself. The exponent applies to only ________ number or symbol unless parenthesis are used. 2x2 = 2(___)(___) = _____ versus (2x)2 = (____)(____) = _____ Evaluate: 1.) 33 2.) 25 3.) -44 5.) -24 6.) (-2)4 7.) 3x2y3 if x = -2 and y = -1 B. Simplifying Zero Exponents Explore: x5 8.) 5 x 9.) 4.) x 2 y8 x 2 y8 10.) 34 34 **For every non-zero number a, a0 = _____** Evaluate: 11.) 60 12.) -30 C. Simplifying Negative Exponents Explore: x3 x2 y6 15.) 5 16.) 3 8 x x y 13.) 6xy0 14.) 17.) 2 40 33 35 1 Algebra 1 Chapter 7 Notes **When I have a ___________ exponent, I make the exponent ___________ and move that term only to the _________________________________.** a-n 1 , a __ 1 a __ -n a Simplify. Express using positive exponents: 18.) x-3 21.) 22 6x 4 19.) 4m-2 20.) 3-2 22.) x-3y4 23.) 3a 2b2 c0 Evaluate each expression if: x = 2 and y = -3 Re-write each expression using positive exponents Plug in values of x and y. Evaluate. 24.) x2y-3 27.) 1 x y3 2 25.) 3-2x-2y2 26.) x-yyx 28.) 2x0y-2 29.) 4x-2 2 Algebra 1 Chapter 7 Notes Name:_______________________ Notes #1: Sections 7.1 – 7.3 Section 7.1: Review Simplify; leave all answers in positive exponents: 1.) m-3 2.) y-1 3.) 6m0 4.) 2-3 5.) 4-2 6.) -3-3 7.) (7m4)0 8.) 11.) a-1 + b-1 12.) (ab)-2 3x 0 22 y 3 Evaluate if a = -2 and b = -3: 9.) a2b-1 10.) 2ab-1 Section 7.2: Scientific Notation A. Why Scientific Notation? Compare: 145,000,000 vs 1.45 x 108 Why do we use scientific notation? 0.00000023 vs 2.3 x 10-7 When will it be used most often? B. Format for Scientific Notation A number in scientific notation is written as the product of two factors in the form a x 10n, where n is an integer and 1 a 10 GOAL: ___ . ___ ___ ___ x 10 ___ Examples: -2.7 x 10-3 3.081 x 1011 Is each number written in scientific notation? If no, explain. 1.) 35 x 107 2.) -3.903 x 101 3.) -12 x 100.5 3 Algebra 1 Chapter 7 Notes C. Writing in Standard Notation Converting from Scientific Notation to Standard Notation • If 10 is to a ________ power, move the decimal that many places to the ________ • If 10 is to a ________ power, move the decimal that many places to the ________ Write in standard notation: -4 4.) 2.35 x 103 5.) -1.4 x 10 6.) 9.876 x 10 -6 7.) -5.61 x 10 8 D. Writing in Scientific Notation Converting from Standard Notation to Scientific Notation • • • • Place the decimal point so that you have 1 digit to the left of the decimal point ___. ___ ___ ___ Count the number of places you need to move the decimal to get back to your original number. This number will be the exponent on the 10. If you are moving to the left, this exponent will be _____________ If you are moving to the right, this exponent will be _____________ Write in scientific notation: 8.) -873 9.) 13,000 8 10.) 0.0063 11.) -0.05 13.) 0.045 x 10-7 14.) 42.301 x 10-3 15.) 0.000375 x 108 16.) 3(1.2 x 109) 17.) 3(4.5 x 105) 18.) 0.5(1.8 x 10-9) 19.) 0.5(12.7 x 103) 20.) 5.2(4.2 x 10-2) 21.) 3.1(0.00004) 22.) 4.5(-5.3 x 108) 23.) (45,000)(123) 12.) 125.7 x 10 4 Algebra 1 Chapter 7 Notes Order the numbers from least to greatest: 24.) 0.052 x 107, 5.12 x 105, 53.2 x 10 and 534 25.) 60.2 x 10-5, 63 x 104, 0.067 x 103, 61 x 10-2 Section 7.3 A. Writing Expressions with Exponents Simplify. Express each in exponential form: 1.) x x x x 4.) x2·x5·x·x3 a a a bbbb 2.) 3 3 3 3 3 3 3 3.) 5.) y3·y2·y4·y 6.) 3·m4·2·m2·m What pattern did you notice with the exponents? B. Multiplying Monomials Multiplying Monomials When I multiply variable expressions together, I _______________ the coefficients AND I ___________ the exponents of terms with the same base. Simplify each expression. Leave answers with positive exponents. 7.) (-3m-2n5)(2mn2) 8.) (-4p3q)(-5p2q-6) 9.) (9x-3y2z)(-3x-4yz) 10.) 35 ∙ 3-2 ∙ 37 11.) 34 ∙ 22 ∙ 3-3 12.) (3c6)(c-2d8)(-2cd-5) C. Applications to Scientific Notation Multiplying Numbers in Scientific Notation Multiply Coefficients (decimal terms) Add Exponents Check to make sure that the answer is in scientific notation 13.) (2.5 x 105)(3.0 x 108) 14.) (1.6 x 10-3)(3.0 x 105) 5 Algebra 1 Chapter 7 Notes 15.) (5.4 x 10-2)(6.2 x 10-6) 16.) (9.3 x 10-4)(3.1 x 105) 17.) (8.4 x 10-7)(3.6 x 10-3) 18.) (-7.3 x 10-4)(9.1 x 108) D. Other Applications Complete each equation (fill-in-the-blank with a number) ___ 6 17.) 3 ___ 32 3 7 8 820 18.) 8 19.) 3x ___ 3x3 9x7 20.) x4y( ) ∙ x( ) = y2 E. Comparing Addition and Multiplication 21.) (4x2)(5x2) vs. 4x2 + 5x2 22.) (7y3)(5y3) 23.) (14b2)(3b3) vs. 14b2 + 3b3 24.) (7n5)(3n4) vs. vs. 7y3 + 5y3 7n5 + 3n4 6 Algebra 1 Chapter 7 Notes Notes #2: Review of Sections 7.1-7.3, Section 7.4 Review of Sections 7.1-7.3 1.) Simplify: 7x-3 4.) Simplify: 2.) Simplify: 4-2 12 x 4 4 x 2 3.) Evaluate 4x-3 if x = -2 5.) Evaluate for a = 3 and b = -2 6.) Convert to scientific notation: 304, 000 6abb0 7.) Convert to scientific notation: 0.0872 8.) Convert to standard notation: 6.2 x 10 4 9.) Convert to scientific notation: 47 x 108 10.) Simplify: (5x3y2)(-3x4y3) 11.) Simplify each: 12.) Multiply; leave in scientific notation: (-5.3 x 10-7)(8.1 x 1012) (9m3)(5m3) 9m3 + 5m3 vs. Section 7.4 A. Raising a Power to a Power Explore: 1.) (x2)4 versus (x2)(x4) 2.) (3y3)2 versus 2(3y2) Raising a Power to a Power Raise the coefficient to the power; evaluate _______________ the exponents on the variables Simplify; leave in terms of positive exponents: 3.) (x3)4 4.) (m4)2(m3)3 5.) (g-2)-3(g4)5 6.) (2w-3)2 7.) (4d2)-3 9.) (p-2)5(p-10) 10.) (3.5)3(3.5)-2 8.) (5x2y4)3(2xy2) 7 Algebra 1 Chapter 7 Notes 11.) (3a2b-4)2(2ab3)3 12.) (35)2(3-2)4 13.) (x-2)(3xy2)4 Complete each equation (fill-in-the-blank with a number) 14.) (m4)___ = m12 15.) m4 ∙ m___ = m12 16.) (a2b4)___ = 1 ab 6 12 B. Applications to Scientific Notation Raise the coefficient to the given power Multiply the exponents Check to make sure that your answer is in scientific notation Simplify. Write each answer in scientific notation 17.) (1.2 x 104)2 18.) (2 x 10-9)-2 19.) (-3 x 105)3 20.) (8 x 10-5)2 Simplify. Leave your answer in scientific notation. 21.) (5.6 x 104)(6.2 x 107) 22.) (3.2 x 106)2 23.) (4.3 x 108)(12.5 x 104) 24.) (3.2 x 10-5)2 25.) 4(6.3 x 107) 26.) (4.3 x 108)(12.5 x 104) Combine like terms: 27.) 7m – 3n + 6n – 8m 28.) 3x2 – 7x + 8x2 – 2x 29.) 3ab4 – 2ab3 + ab4 + 8ab2 30.) 6j3k2 – 7j3k2 + 8jk – jk 31.) 19w8z3 – 8w7z2 + 11w8z3 32.) 10x4y3z – 2x4y3z2 + 4x4y3z 8 Algebra 1 Chapter 7 Notes Review Topics: 1.) Multiply. Leave your answer in scientific notation: 6.2 x 104 3.1 x 1011 3.) Are the lines parallel, perpendicular, or neither? y 3x 1 2.) Find f(-3) if f(x) = 4 – 2x2 4.) Find the x-intercept and the y-intercept of the line: 2x – 7y = 10 3x 2 y 7 5.) Solve: 2 x 1 5 6.) Solve for x and y: 4 x 2 y 16 11x 3 y 7 7.) Graph the linear inequality: y 10 9 3x 2 y 10 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 9 Algebra 1 Chapter 7 Notes Notes #3: Review and Section 7.5 Review: Combine like terms. Distribute/Multiply first if necessary. 1.) x 2 3x 2 x 2 x 2.) 2 y 2 6 y 3 y 2 4 y 9 y 5 3.) 3x(2x – 5) + 4x(8 – 3x) 4.) -7x2(6x – 2) + 3x(8 – 2x2) 6.) (4x3)2 – (5x4)(6x2) 5.) 5x(3x – 2y) – 2x(3y – 7x) 7.) (3a3)(-4a5) + (2a2)(5a4) + (3a2)4 – (2a3)2 Review Topics: 8.) Multiply; leave in scientific notation: (8.2 x 108)(6.1 x 10-3) 9.) Evaluate; leave in scientific notation: (4.3 x 105)2 10.) Simplify: (3a3b6)(-5a4b-9) 11.) Simplify: (3w3z8)2(-2wz2) 6m 2 n 5 12.) Simplify: 1 2 2m n 13.) Simplify each: 3 (-4b6)(5b6) -4b6 + 5b6 vs. Section 7.5: Division Properties of Exponents A. Dividing Monomials Explore: a6 1.) 2 a 2.) x4 y3 x2 y 3.) 57 53 10 Algebra 1 Chapter 7 Notes Dividing Monomials **When I divide like numbers or variables, I ___________ their exponents.** am a mn n a Use this property to simplify the expressions. Leave all answers in positive exponents. 65 x 4 73 m 7 g6 4.) 6 5.) 6.) 62x 7 m5 g x5 y 4 z 2 8.) 3 4 x y z c 2 d 5 7.) 9 7 cd 4 1 10.) 4 3 4 4 x 2 y 3 5 x 2 z 8 9.) 6 8 5 z 2 y 2 m 2 12.) 4 3n x5 y 3 z 6 11.) 12 13 16 x y z 3 B. Fractions to Negative Powers 2 2 ___ 3___ 2 Explore: ___ ___ 3 2 3 2 This is the same as: 2 3 2 Fractions to Negative Powers **When I take a fraction to a negative power, I ___________ the fraction over, and raise it to the ____________________ power. ** a b 2 13.) 5 2 4 14.) 3 3 n b a n 4 x 15.) 5 2 3 w 4 16.) 3 2y 2 11 Algebra 1 Chapter 7 Notes C. Applications to Scientific Notation Dividing Numbers in Scientific Notation Divide Coefficients (decimal terms) Subtract Exponents Check to make sure that the answer is in scientific notation 17.) 4.8 x 108 2.4 x 102 18.) 4.8 x 105 6.0 x 102 19.) 1.2 x 1010 3.0 x 104 4.2 x 10 21.) 5 2 6.9 x 105 20.) 2.3 x 104 2.1 x 104 Review Topics: 22.) Solve for x and y: 23.) Solve for x and y: 8 x 2 y 2 4 x 5 y 15 y 5 x 1 6 x 4 y 11 24.) Graph the line: 4x – y = -8 25.) Find the equation of the line with slope 2 and passing through the point (-6, 5). Leave 3 your answer in slope intercept form. y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 12 Algebra 1 Chapter 7 Notes Notes #4: Chapter 7 Review Evaluate: 1.) 3 4 2 2.) -33 5.) -4-2 9.) 6.) 28m8 n3 21m 2 n7 3.) (-3)4 32 x0 y 2 61 x5 y 3 4.) -5(6x)0 7.) (-4a-5b7) (-3a4b-3)2 10.) (-2b3c5)(-5b2c)(b2c) 8.) (t6)4 11.) (-4w-3)-2(6w5) Write in standard notation: 12.) -4.602 x 10-5 13.) 7.041 x 107 Write in scientific notation 14.) 0.00354 15.) 8,900,000 3 2 17.) (5.0 x 10 ) 16.) (3 x 103)(0.2 x 10-2) 1.2 x 107 18.) 4.0 x 102 Simplify: 19.) 32 2 3 2 2 3x 2 y 3 4 x3 z 8 20.) 8 6 5z y 1 13 Algebra 1 Chapter 7 Notes Review Topics: 1.) Divide. Leave your answer in scientific notation: 3.6 x 1014 2.) Find h(-1) if h(x) = 6x – 3 6.0 x 10 3 3.) Does the system have no solution, one solution, or infinitely many solutions? 4.) Simplify: 8m3 40m2 12m4 60m3 y 2x 3 4x 2 y 6 5.) Find the equation of the line with slope ½ and passing through the point (-4, 7) 7.) Simplify: (4x – y)2 6.) Find the equation of the line passing through the points (3, 0) and (1, 10). 8.) Find g(-1) if g(x) = 5 – 3x 14 Algebra 1 Chapter 7 Notes 9.) Find the slope and y-intercept of each line. What can you conclude? 10.) Graph: x 4 y 7 y 5 x 3 y 12 3 3 x y 5 5 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 11.) Find the equation of the line that is parallel to y = -2x + 1 and passing through the point (5, 2) 12.) Find the equation of the line that is perpendicular to y = -2x + 1 and passing through the point (5, 2) 15 Algebra 1 Chapter 7 Notes Algebra 1: Chapter 7 Study Guide For #1-8, simplify each expression: 1.) (-3)0 2.) (4)-2 3.) -2-3 5.) (7m4)0 7.) 3-2w3x-3y-2 6.) (2f)-4 Name: __________________ 4.) 1 50 4ab 3 8.) 1 2 2 c For #9-11, evaluate each expression for m = -2 and n = 3 9.) m-2n 10.) 2-3m3n-2 11.) m-n For #12-14, write each number in scientific notation: 12.) 0.0065 13.) 130,000 14.) 0.03025 For #15-17, write each number in standard notation: 15.) 4.3 x 10-3 16.) 2.75 x 107 17.) 8 x 10-5 For #18-20, is each number written in scientific notation? If not, convert to scientific notation. 18.) 44 x 103 19.) 0.5 x 10-4 20.) 3.45 x 10-2 For #21-23, rewrite each expression using the base only once 21.) (23)(25)(2-2) 58 22.) 2 5 23.) (9.7)-8(9.7)8 16 Algebra 1 Chapter 7 Notes For #24-35, simplify each expression. If applicable, leave your answer in scientific notation. 24.) (3bc8)(-5b2c)(2b3c-2) 25.) (8.5 x 107)(-1.2 x 10-3) 26.) (d3)2(d2)4 27.) (-5m-2n3)3 28.) (2 x 103)2 29.) (-3x2y3)2(-8xy5) 2 30.) 27m7 n5 3m9 n 4 xy 2 9 xz 2 31.) 3 3z 2 y 33.) 3.6 x 105 1.2 x 102 34.) 3t 1 32.) 3 4r 3.5 x 106 7 x 104 2 5 j 2 35.) 24 j 3k 5 3k For #36-39, complete each equation (fill in the blank). 36.) 6_ 65 62 37.) 3x_ 2x3 6x7 38.) ab_ a _ a5b2 39.) 8385 _ 8 2 8 8 17 Algebra 1 Chapter 7 Notes 40.) Simplify: 6m2 3m 8 4m2 2m 1 41.) Simplify: 3x(4 xy 5 y 2 ) 7 y 2 x 8 x 2 42.) Find the slope of the line passing through the points: (-4, 6) and (1, -4) 43.) Find the equation of the line with slope of 1 that passes through the point (9, -4). Leave 2 in slope-intercept form. 44.) Graph the line: 2x – y = -6 45.) Solve for x: x 3 5 7 y 10 9 8 7 6 5 4 3 2 1 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 9 10 46.) Solve for x: 3x 1 7 -2 -3 -4 -5 -6 -7 -8 -9 -10 47.) Solve for x and y: 3 x y 13 x y5 49.) Are the lines parallel, perpendicular, or neither? y 2x 5 2x y 5 48.) Find g(-2) if g(x) = 5 – 3x2 50.) Find the x-intercept and the y-intercept of the line: 3x 4 y 9 18