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Mastering Exponents: Properties, Products, and Quotients

Delve into the fascinating world of exponents and explore their critical properties in mathematics. This resource covers the fundamentals of powers, including base and exponent definitions, succinctly explaining concepts like the Product of Powers, Power of a Power, and Power of a Product properties. Gain insight into simplifying expressions involving exponents, as well as applying these properties to products and quotients. Understand zero and negative exponents and practice through guided examples. Perfect for students wanting to strengthen their algebra skills!

alfonso-mccarthy
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Mastering Exponents: Properties, Products, and Quotients

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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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