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Zero and Negative Exponents

Zero and Negative Exponents. Lesson 8-1. Objectives: 1. to simplify expressions with zero and negative exponents 2. to evaluate exponential expressions. Use the definition of negative exponent. 1 3 2. =. Simplify. 1 9. =. = 1.

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Zero and Negative Exponents

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  1. Zero and Negative Exponents Lesson 8-1 Objectives: 1. to simplify expressions with zero and negative exponents 2. to evaluate exponential expressions

  2. Use the definition of negative exponent. 1 32 = Simplify. 1 9 = = 1 Use the definition of zero as an exponent. Zero and Negative Exponents Lesson 8-1 Simplify. a. 3–2 (–22.4)0 b.

  3. 1 x–3 = 1  x–3 1 b2 = 3a Rewrite using a division symbol. Use the definition of negative exponent. 3a b2 = Use the definition of negative exponent. 1 x3 = 1  Simplify. = 1 • x3 1 x3 Multiply by the reciprocal of , which is x3. = x3 Identity Property of Multiplication Zero and Negative Exponents Lesson 8-1 Simplify a. b. 3ab–2

  4. 4x2 y3 Use the definition of negative exponent. 4x2y–3 = Substitute 3 for x and –2 for y. = 4(3)2 Simplify. (–2)3 36 –8 1 2 = = –4 Zero and Negative Exponents Lesson 8-1 Evaluate 4x2y–3 for x = 3 and y = –2. Method 1:Write with positive exponents first.

  5. 4x2y–3 = 4(3)2(–2)–3 Substitute 3 for x and –2 for y. Use the definition of negative exponent. 4(3)2 (–2)3 = Simplify. 36 –8 1 2 = = –4 Zero and Negative Exponents Lesson 8-1 (continued) Method 2: Substitute first.

  6. Zero and Negative Exponents Lesson 8-1 In the lab, the population of a certain bacteria doubles every month. The expression 3000 • 2m models a population of 3000 bacteria after m months of growth. Evaluate the expression for m = 0 and m = –2. Describe what the value of the expression represents in each situation. a. Evaluate the expression for m = 0. 3000 • 2m = 3000 • 20Substitute 0 for m. = 3000 • 1 Simplify. = 3000 When m = 0, the value of the expression is 3000. This represents the initial population of the bacteria. This makes sense because when m = 0, no time has passed.

  7. b. Evaluate the expression for m = –2. 3000 • 2m = 3000 • 2–2Substitute –2 for m. = 3000 • Simplify. = 750 1 4 Zero and Negative Exponents Lesson 8-1 (continued) When m = –2, the value of the expression is 750. This represents the 750 bacteria in the population 2 months before the present population of 3000 bacteria.

  8. Scientific Notation Lesson 8-2 Objectives: 1. to write numbers in scientific and standard notation 2. to use scientific notation

  9. Scientific Notation Lesson 8-2 Is each number written in scientific notation? If not, explain. a. 0.46  104 No; 0.46 is less than 1. b. 3.25  10–2 yes c. 13.2  106 No; 13.2 is greater than 10.

  10. Move the decimal point 8 places to the left and use 8 as an exponent. 2.34  108 234,000,000 = Drop the zeros after the 4. Move the decimal point 5 places to the right and use –5 as an exponent. 6.3  10–5 0.000063 = Drop the zeros before the 6. Scientific Notation Lesson 8-2 Write each number in scientific notation. a. 234,000,000 b. 0.000063

  11. A positive exponent indicates a number greater than 10. Move the decimal point 4 places to the right. 8.8  104 = 8.8000. A negative exponent indicates a number between 0 and 1. Move the decimal point 5 places to the left. 7.3  10–5 = 0.00007.3 Scientific Notation Lesson 8-2 Write each number in standard notation. a. elephant’s mass: 8.8  104 kg = 88,000 b. ant’s mass: 7.3  10–5 kg   = 0.000073

  12. Distance from the Sun Planet 4.84  108 mi Jupiter 9.3  107 mi Earth 4.5  109 mi Neptune 3.8  107 mi Mercury Scientific Notation Lesson 8-2 List the planets in order from least to greatest distance from the sun. Order the powers of 10. Arrange the decimals with the same power of 10 in order.

  13. Distance from the Sun Planet 4.84  108 mi Jupiter 9.3  107 mi Earth 4.5  109 mi Neptune 3.8  107 mi Mercury Scientific Notation Lesson 8-2 (continued) 3.8  107 9.3  107 4.84  108 4.5  109 Mercury Earth Jupiter Neptune From least to greatest distance from the sun, the order of the planets isMercury, Earth, Jupiter, and Neptune.

  14. Write each number in scientific notation. 0.0063 105 6.03  104 6103 63.1  103 6.3  102 6.03  104 6.103  103 6.31  104 Scientific Notation Lesson 8-2 Order 0.0063  105, 6.03  104, 6103, and 63.1  103 from least to greatest. Order the powers of 10. Arrange the decimals with the same power of 10 in order. 6.3  102 6.103  103 6.03  104 6.31  104 Write the original numbers in order. 0.0063  105 6103 6.03  104 63.1  103

  15. Use the Associative Property of Multiplication. (6 • 8)  10–4 = = 48 10–4 Simplify inside the parentheses. = 4.8  10–3 Write the product in scientific notation. Use the Associative Property of Multiplication. (0.3 • 1.3)  103 = = 0.39 103 Simplify inside the parentheses. = 3.9  102 Write the product in scientific notation. Scientific Notation Lesson 8-2 Simplify. Write each answer using scientific notation. 6(8 10–4) a. 0.3(1.3 103) b.

  16. Multiplication Properties of Exponents Lesson 8-3 Objectives: 1. to multiply powers, with the same base 2. to work with scientific notation

  17. Add exponents of powers with the same base. = 73 + 2 = 75 Simplify the sum of the exponents. Think of 4 + 1 – 2 as 4 + 1 + (–2) to add the exponents. 44 + 1 – 2 = = 43 Simplify the sum of the exponents. Add exponents of powers with the same base. 68 + (–8) = = 60 Simplify the sum of the exponents. = 1 Use the definition of zero as an exponent. Multiplication Properties of Exponents Lesson 8-3 Rewrite each expression using each base only once. a. 73 • 72 b. 44 • 41 • 4–2 c. 68 • 6–8

  18. b. 2q • 3p3 • 4q4 p2 + 1 + 5 Add exponents of powers with the same base. = = p8 Simplify. = (2 • 3 • 4)(p3)(q • q4) Commutative and Associative Properties of Multiplication = 24(p3)(q1• q 4) = 24(p3)(q1 +q 4) Multiply the coefficients. Write q as q1. Add exponents of powers with the same base. = 24p3q5 Simplify. Multiplication Properties of Exponents Lesson 8-3 Simplify each expression. a. p2 • p • p5

  19. Commutative and Associative Properties of Multiplication (3  10–3)(7  10–5)= (3 • 7)(10–3• 10–5) = 21 10–8 Simplify. = 2.1  101 • 10–8 Write 21 in scientific notation. Add exponents of powers with the same base. = 2.1  101+ (– 8) = 2.1 10–7 Simplify. Multiplication Properties of Exponents Lesson 8-3 Simplify (3  10–3)(7  10–5). Write the answer in scientific notation.

  20. meters seconds kilometers meters seconds hour Use dimensional analysis. Speed of light = • • Substitute. m s = (3  108) • (1  10–3) • (3.6  103) Commutative and Associative Properties of Multiplication = (3 • 1 • 3.6)  (108 • 10–3 • 103) s h km m = 10.8 (108+ (– 3)+ 3) Simplify. Multiplication Properties of Exponents Lesson 8-3 The speed of light is 3  108 m/s. If there are 1 10–3 km in 1 m, and 3.6  103 s in 1 h, find the speed of light in km/h.

  21. = 10.8  108 Add exponents. = 1.08  101 • 108 Write 10.8 in scientific notation. = 1.08  109 Add the exponents. Multiplication Properties of Exponents Lesson 8-3 (continued) The speed of light is about 1.08  109 km/h.

  22. More Multiplication Properties of Exponents Lesson 8-4 Objectives: 1. to raise a power to a power 2. to raise a product to a power

  23. Multiply exponents when raising a power to a power. (a3)4 = a3• 4 Simplify. = a12 More Multiplication Properties of Exponents Lesson 8-4 Simplify (a3)4.

  24. b2(b3)–2 = b2 • b3• (–2) Multiply exponents in (b3)–2. = b2 • b–6 Simplify. Add exponents when multiplying powers of the same base. = b2 + (–6) = b–4 Simplify. 1 b4 Write using only positive exponents. = More Multiplication Properties of Exponents Lesson 8-4 Simplify b2(b3)–2.

  25. (4x3)2 = 42(x3)2 Raise each factor to the second power. Multiply exponents of a power raised to a power. = 42x6 = 16x6 Simplify. More Multiplication Properties of Exponents Lesson 8-4 Simplify (4x3)2.

  26. Raise the three factors to the second power. (4xy3)2(x3)–3 = 42x2(y3)2 • (x3)–3 Multiply exponents of a power raised to a power. = 42 • x2 • y6 • x–9 Use the Commutative Property of Multiplication. = 42 • x2 • x–9 • y6 Add exponents of powers with the same base. = 42 • x–7 • y6 16y6 x7 Simplify. = More Multiplication Properties of Exponents Lesson 8-4 Simplify (4xy3)2(x3)–3.

  27. Raise each factor within parentheses to the second power. 102 • (3 108)2 = 102 • 32 • (108)2 = 102 • 32 • 1016 Simplify (108)2. Use the Commutative Property of Multiplication. = 32 • 102 • 1016 Add exponents of powers with the same base. = 32 • 102 + 16 Simplify. Write in scientific notation. = 9  1018 More Multiplication Properties of Exponents Lesson 8-4 An object has a mass of 102 kg. The expression 102 • (3  108)2 describes the amount of resting energy in joules the object contains. Simplify the expression.

  28. Division Properties of Exponents Lesson 8-5 Objectives: 1. to divide powers with the same base 2. to raise a quotient to a power

  29. Subtract exponents when dividing powers with the same base. x4 – 9 = x4 x9 = x–5 Simplify the exponents. Rewrite using positive exponents. 1 x5 = p3j –4 p–3j 6 Subtract exponents when dividing powers with the same base. = p3 – (–3)j –4 – 6 = p6 j –10 Simplify. Rewrite using positive exponents. p6 j10 = Division Properties of Exponents Lesson 8-5 Simplify each expression. a. b.

  30. 64 million beats 530 thousand min = 6.4  107 beats Write in scientific notation. 5.3  105 min Subtract exponents when dividing powers with the same base. 6.4 5.3 Simplify the exponent. 107–5 = 6.4 5.3 102 = 1.21  102 Divide. Round to the nearest hundredth. = 121 Write in standard notation. Division Properties of Exponents Lesson 8-5 A small dog’s heart beats about 64 million beats in a year. If there are about 530 thousand minutes in a year, what is its average heart rate in beats per minute? The dog’s average heart rate is about 121 beats per minute.

  31. Raise the numerator and the denominator to the fourth power. 3 y3 4 = 34 (y3)4 Multiply the exponent in the denominator. Simplify. 34 y12 = 81 y12 = Division Properties of Exponents Lesson 8-5 3 y3 4 Simplify .

  32. 2 3 Rewrite using the reciprocal of . 2 3 3 2 –3 3 = Raise the numerator and the denominator to the third power. Simplify. = 3 8 3 27 8 or 33 23 = Division Properties of Exponents Lesson 8-5 2 3 –3 a. Simplify .

  33. 2 –2 Rewrite using the reciprocal of . = Write the fraction with a negative numerator. 2 = Raise the numerator and denominator to the second power. (–c)2 (4b)2 = c2 16b2 4b c 4b c 4b c c 4b c 4b – – – – – Simplify. = Division Properties of Exponents Lesson 8-5 (continued) –2 . b. Simplify

  34. Geometric Sequences Lesson 8-6 Objectives: 1. to form geometric sequences 2. to use equations/rules when describing geometric sequences

  35. 3 –15 75 –375 (–5) (–5) (–5) 3 2 3 4 3 8 1 2 1 2 1 2 b. 3, , , , ...    1 2 3 2 3 4 3 8 3 The common ratio is . Geometric Sequences Lesson 8-6 Find the common ratio of each sequence. a. 3, –15, 75, –375, . . . The common ratio is –5.

  36. 5 –10 20 –40 (–2) (–2) (–2) Geometric Sequences Lesson 8-6 Find the next three terms of the sequence 5, –10, 20, –40, . . . The common ratio is –2. The next three terms are –40(–2) = 80, 80(–2) = –160, and –160(–2) = 320.

  37. 62 54 18 6 1 3 1 3 1 3    Geometric Sequences Lesson 8-6 Determine whether each sequence is arithmetic or geometric. a. 162, 54, 18, 6, . . . The sequence has a common ratio. The sequence is geometric.

  38. 98 101 104 107 + 3 + 3 + 3 Geometric Sequences Lesson 8-6 (continued) b. 98, 101, 104, 107, . . . The sequence has a common difference. The sequence is arithmetic.

  39. Geometric Sequences Lesson 8-6 Find the first, fifth, and tenth terms of the sequence that has the rule A(n) = –3(2)n – 1. first term: A(1) = –3(2)1– 1 = –3(2)0 = –3(1) = –3 fifth term: A(5) = –3(2)5 – 1 = –3(2)4 = –3(16) = –48 tenth term: A(10) = –3(2)10 – 1 = –3(2)9 = –3(512) = –1536

  40. The first term is 2 meters, which is 200 cm. Draw a diagram to help understand the problem. Geometric Sequences Lesson 8-6 Suppose you drop a tennis ball from a height of 2 meters. On each bounce, the ball reaches a height that is 75% of its previous height. Write a rule for the height the ball reaches on each bounce. In centimeters, what height will the ball reach on its third bounce?

  41. A rule for the sequence is A(n) = 200 • 0.75n– 1. Use the sequence to find the height of the third bounce. A(n) = 200 • 0.75n – 1 Substitute 4 for n to find the height of the third bounce. A(4) = 200 • 0.754 – 1 = 200 • 0.753 Simplify exponents. = 200 • 0.421875 Evaluate powers. = 84.375 Simplify. Geometric Sequences Lesson 8-6 (continued) The ball drops from an initial height, for which there is no bounce. The initial height is 200 cm, when n = 1. The third bounce is n = 4. The common ratio is 75%, or 0.75. The height of the third bounce is 84.375 cm.

  42. Exponential Functions Lesson 8-7 Objectives: 1. to evaluate exponential functions 2. to graph exponential functions

  43. 1 16 3 16 3 16 –2 3 • 4–2 = 3 • = Exponential Functions Lesson 8-7 Evaluate each exponential function. a.y = 3x for x = 2, 3, 4 xy = 3xy 2 32 = 9 9 3 33 = 27 27 4 34 = 81 81 b.p(q) = 3 • 4q for the domain {–2, 3} qp(q) = 3 • 4qp(q) 3 3 • 43 = 3 • 64 = 192 192

  44. In two years, there are 8 three-month time periods. ƒ(x) = 2 • 48 ƒ(x) = 2 • 65,536 Simplify powers. ƒ(x) = 131,072 Simplify. Exponential Functions Lesson 8-7 Suppose two mice live in a barn. If the number of mice quadruples every 3 months, how many mice will be in the barn after 2 years? ƒ(x) = 2 • 4x

  45. 2 9 2 3 2 2 9 –2 2 • 3–2 = = (–2, ) 2 31 2 3 2 3 –1 2 • 3–1 = = (–1, ) 0 2 • 30 = 2 • 1 = 2 (0, 2) 1 2 • 31 = 2 • 3 = 6 (1, 6) 2 2 • 32 = 2 • 9 = 18 (2, 18) Exponential Functions Lesson 8-7 Graph y = 2 • 3x. xy = 2 • 3x (x, y)

  46. 1 1.251 = 1.25 1.3 (1, 1.3) 2 1.252 = 1.5625 1.6 (2, 1.6) 3 1.253 = 1.9531 2.0 (3, 2.0) 4 1.254 = 2.4414 2.4 (4, 2.4) 5 1.255 = 3.0518 3.1 (5, 3.1) Exponential Functions Lesson 8-7 The function ƒ(x) = 1.25x models the increase in size of an image being copied over and over at 125% on a photocopier. Graph the function. x ƒ(x) = 1.25x (x, ƒ(x))

  47. Exponential Growth and Decay Lesson 8-8 Objectives: 1. to model exponential growth 2. to model exponential decay

  48. Relate:  y = a • bx Use an exponential function. Define: Let x = the number of years since 1998. Let y = the population of the town at various times. Let a = the initial population in 1998, 13,000 people. Let b = the growth factor, which is 100% + 1.4% = 101.4% = 1.014. Exponential Growth and Decay Lesson 8-8 In 1998, a certain town had a population of about 13,000 people. Since 1998, the population has increased about 1.4% a year. a. Write an equation to model the population increase. Write: y = 13,000 • 1.014x

  49. 2006 is 8 years after 1998, so substitute 8 for x. y = 13,000 • 1.0148 Use a calculator. Round to the nearest whole number. 14,529 Exponential Growth and Decay Lesson 8-8 (continued) b. Use your equation to find the approximate population in 2006. y = 13,000 • 1.014x The approximate population of the town in 2006 is 14,529 people.

  50. Relate:  y = a • bx Use an exponential function. Define: Let x = the number of interest periods. Let y = the balance. Let a = the initial deposit, $1000 Let b = 100% + 7.2% = 107.2% = 1.072. Once a year for 5 years is 5 interest periods. Substitute 5 for x. = 1000 • 1.0725 Use a calculator. Round to the nearest cent. 1415.71 Exponential Growth and Decay Lesson 8-8 Suppose you deposit $1000 in a college fund that pays 7.2% interest compounded annually. Find the account balance after 5 years. Write: y = 1000 • 1.072x The balance after 5 years will be $1415.71.

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