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Objective :

Objective :. To apply the properties of exponents. Properties of Exponents. Power: A power is an expression that represents repeated multiplication of the same factor.

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Objective :

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  1. Objective: • To apply the properties of exponents.

  2. PropertiesofExponents • Power: A power is an expression that represents repeated multiplication of the same factor. • For example, 81 is a power of 3 because 3x3x3x3 = 81. A power can be written using two numbers, a base and an exponent. • Exponent: The exponent represents the number of times the base is used as a factor. • Base: “The big number”

  3. Lesson 8.1 • Apply Exponent Properties Involving Products

  4. *Power # 1 Product of Powers Property:

  5. a. = 78 73 75 = 73 + 5 b. 9 98 92 = 91 98 92 x4x3 = (– 5)1 (– 5)6 c. (– 5)(– 5)6 = x4 + 3 = x7 d. Use the product of powers property EXAMPLE 1 = 91 + 8 + 2 = 911 = (– 5)1 + 6 = (–5)7

  6. 32 37 1. 2. 5 59 3. (– 7)2(– 7) 4. x2x6x for Example 1 GUIDED PRACTICE Simplify the expression. = 39 = 510 = (–7)3 = x9

  7. Reading Math 5 4 (9 ) is read as “nine to the fourth, to the fifth.” *Power # 2 Power of a Power Property:

  8. [(–6)2]5 a. b. (25)3 = (–6)2 5 = 253 (x2)4 [(y + 2)6]2 c. d. = x24 EXAMPLE 2 Use the power of a power property = 215 = (–6)10 = (y + 2)6 2 = (y + 2)12 = x8

  9. [(–2)4]5 6. 5. (42)7 (n3)6 [(m + 1)5]4 7. 8. for Example 2 GUIDED PRACTICE Simplify the expression. = 414 = (–2)20 = n18 = (m + 1)20

  10. *Power # 3 Power of a Product Property: • To find a power of a product, find the power of each factor and multiply.

  11. (24 13)8 = 248 138 a. b. c. d. (–4z)2 = (–4 z)2 = (–4)2z2 = 16z2 (9xy)2 = (9 x y)2 = 92x2y2 = 81x2y2 – (4z)2 = – (4 z)2 = – (42z2) = –16z2 EXAMPLE 3 Use the power of a product property

  12. (2x3)2x4 = 22 (x3)2x4 = 4 x6x4 (2x3)2x4 EXAMPLE 4 Use all three properties Simplify Power of a product property Power of a power property = 4x10 Product of powers property

  13. (42  12)2 9. (–3n)2 10. 11. (9m3n)4 5  (5x2)4 12. for Examples 3, 4 and 5 GUIDED PRACTICE Simplify the expression. = 422 122 = 9n2 = 6561m12n4 = 3125x8

  14. Lesson 8.2 • Apply Exponent Properties Involving Quotients

  15. *Power # 4 Quotient of Powers Property

  16. a. (– 3)9 b. (– 3)3 512 810 84 57 54 58 = c. 57 EXAMPLE 1 Use the quotient of powers property = 810– 4 = 86 = (– 3)9 – 3 = (– 3)6 = 512 – 7 = 55

  17. x6 d. x6 = x4 1 x4 EXAMPLE 1 Use the quotient of powers property = x6 – 4 = x2

  18. 1. 4. y8 (– 4)9 2. (– 4)2 1 611 65 y5 94 93 92 3. for Example 1 GUIDED PRACTICE Simplify the expression. = 66 = (– 4)7 = 95 = y3

  19. *Power # 5 Power of a Quotient Property • To find a power of a quotient, find the power of the numerator and the power of the denominator and divide.

  20. 3 a. = – 7 (– 7)2 49 x x2 x2 x 7 x3 2 2 x y y3 – = = = b. EXAMPLE 2 Use the power of quotient property

  21. 3 4x2 64x6 (4x2)3 a. = a8 5y 125y3 (5y)3 = 2b5 43 (x2)3 = 53y3 = 1 1 a10 a2 (a2)5 2a2 b5 2a2 1 b 5 b. = 2a2 b5 = a10 2a2b5 = EXAMPLE 3 Use properties of exponents Power of a quotient property Power of a product property Power of a power property Power of a quotient property Power of a power property Multiply fractions. Quotient of powers property

  22. x2 4y 8. 3t 16 2 5. = 125 y3 3 – – = 6. 5 a2 a 3 2s t5 y b2 b 2 7. x4 = s3 t2 16y2 54 = for Examples 2 and 3 GUIDED PRACTICE Simplify the expression.

  23. Lesson 8.3 • Define and use Zero and negative exponents

  24. *Power # 6 Definition of zero and negative exponents • Anything to the power of zero is 1 50= 1 • a-n is the reciprocal of an. 2-1= ½ • an is the reciprocal of a-n. 2= 1/2 -1

  25. 1 1 = 9 32 = EXAMPLE 1 Use definition of zero and negative exponents a. 3– 2 Definition of negative exponents Evaluate exponent. b. (–7)0 = 1 Definition of zero exponent

  26. 1 0 5 = (Undefined) = 1 1 1 –2 1 c. = 25 1 2 5 5 EXAMPLE 1 Use definition of zero and negative exponents Definition of negative exponents Evaluate exponent. = 25 Simplify by multiplying numerator and denominator by25. d. 0 – 5 a –nis defined only for a nonzero number a.

  27. 0 2 1. 1 3. 1 = 3 2 –3 1 = 64 for Example 1 GUIDED PRACTICE Evaluate the expression. = 8 4. (–1 )0 = 1 2. (–8) – 2

  28. Lesson 8.1 – 8.3 • All of the properties of exponents can be used together!

  29. a. 6– 4 64 EXAMPLE 2 Evaluate exponential expressions = 6– 4 + 4 Product of a power property = 60 Add exponents. = 1 Definition of zero exponent

  30. 1 1 = 256 3– 4 c. 1 = 4 4 EXAMPLE 2 Evaluate exponential expressions b. (4– 2)2 = 4– 2 ∙ 2 Power of a power property = 4– 4 Multiply exponents. Definition of negative exponents Evaluate power. = 34 Definition of negative exponents = 81 Evaluate power.

  31. 1 1 = 125 53 d. 5– 1 52 = EXAMPLE 2 Evaluate exponential expressions = 5– 1– 2 Quotient of powers property = 5– 3 Subtract exponents. Definition of negative exponents Evaluate power.

  32. 1 4– 3 1 ) 7. (– 3 (– 3 ) – 5 5. 5 = 1296 6– 2 8. 62 for Example 2 GUIDED PRACTICE Evaluate the expression. = 64 = 1 6. (5– 3) –1 = 125

  33. = 23x3 (y–5)3 = 8 x3y–15 8x3 = y15 EXAMPLE 3 Use properties of exponents Simplify the expression. Write your answer using only positive exponents. a. (2xy–5)3 Power of a product property Power of a power property Definition of negative exponents

  34. b. (2x)–2y5 –4x2y2 y5 y5 y5 = –16x4y2 (2x)2(–4x2y2) (4x)2(–4x2y2) y3 – = 16x4 = = EXAMPLE 3 Use properties of exponents Definition of negative exponents Power of a product property Product of powers property Quotient of powers property

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