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Columbus State Community College

Columbus State Community College. Chapter 8 Section 1 The Product Rule and Power Rules for Exponents. m. a. a m. b. b m. The Product Rule and Power Rules for Exponents. Review the use of exponents. Use the product rule for exponents. Use the exponent rule ( a m ) n = a m n .

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Columbus State Community College

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  1. Columbus State Community College Chapter 8 Section 1 The Product Rule and Power Rules for Exponents

  2. m a a m b b m The Product Rule and Power Rules for Exponents • Review the use of exponents. • Use the product rule for exponents. • Use the exponent rule ( a m ) n = a m n. • Use the exponent rule ( a b ) m = a m b m. • Use the exponent rule = .

  3. Review of Using Exponents EXAMPLE 1 Review of Using Exponents Write 5 • 5 • 5 • 5 in exponential form, and find the value of the exponential expression. Since 5 appears as a factor 4 times, the base is 5 and the exponent is 4. Writing in exponential form, we have 54. 54 = 5 • 5 • 5 • 5 = 625

  4. Evaluating Exponential Expressions EXAMPLE 2 Evaluating Exponential Expressions Evaluate each exponential expression. Name the base and the exponent. Base Exponent ( a )24 = 2 • 2 • 2 • 2 = 16 2 4 ( b )–24 = – ( 2 • 2 • 2 • 2 ) = – 16 2 4 ( c ) ( –2 )4 = ( – 2 )( – 2 )( – 2 )( – 2 ) = 16 –2 4

  5. Understanding the Base CAUTION It is important to understand the difference between parts (b) and (c) of Example 2. In – 24 the lack of parentheses shows that the exponent 4 applies only to the base 2. In ( – 2 )4 the parentheses show that the exponent 4 applies to the base – 2. In summary, – a m and ( – a )mmean different things. The exponent applies only to what is immediately to the left of it. Expression Base Exponent Example – an a n –52 = –( 5 • 5 ) = – 25 (– a )n – a n (–5 )2= (–5 ) (–5 ) = 25

  6. Product Rule for Exponents Product Rule for Exponents If m and n are positive integers, then am • an = am + n(Keep the same base and add the exponents.) Example: 34• 32 = 34 + 2 = 36

  7. Common Error Using the Product Rule CAUTION Avoid the common error of multiplying the bases when using the product rule. Keep the same base and add the exponents. 34 • 32≠ 96 34 • 32 = 36

  8. Using the Product Rule EXAMPLE 3 Using the Product Rule Use the product rule for exponents to find each product, if possible. ( a ) 62 • 67 = 6 2 + 7 = 6 9 by the product rule. ( b ) (–7 )1(–7 )5 ( b ) (– 7)1(– 7)5 = (– 7)1 + 5 = (– 7)6by the product rule. ( c )47• 32 The product rule doesn’t apply. The bases are different. ( d )x9 • x5 = x9 + 5 = x14 by the product rule.

  9. Using the Product Rule EXAMPLE 3 Using the Product Rule Use the product rule for exponents to find each product, if possible. ( e ) 82+83 The product rule doesn’t apply because this is a sum. ( f ) ( 5 mn4) ( – 8m6n11) = ( 5 •– 8 ) • ( mm6 ) • ( n4n11 ) using the commutative and associative properties. = – 40m7n15by the product rule.

  10. Product Rule and Bases CAUTION The bases must be the same before we can apply the product rule for exponents.

  11. Understanding Differences in Exponential Expressions CAUTION Be sure you understand the difference between adding and multiplying exponential expressions. Here is a comparison. Adding expressions 3 x4+ 2x4 =5x4 Multiplying expressions ( 3 x4) ( 2x5) =6x9

  12. Power Rule (a) for Exponents Power Rule (a) for Exponents If m and n are positive integers, then ( am )n = am n(Raise a power to a power by multiplying exponents.) Example: ( 35)2 = 35•2 = 310

  13. Using Power Rule (a) EXAMPLE 4 Using Power Rule (a) Use power rule (a) to simplify each expression. Write answers in exponential form. ( a ) ( 32)7 = 32•7 = 314 ( b ) ( 65)9 = 65•9 = 645 ( c ) ( w4)2 = w4•2 = w8

  14. Power Rule (b) for Exponents Power Rule (b) for Exponents If m is a positive integer, then ( a b )m = am bm(Raise a product to a power by raising each factor to the power.) Example: ( 5a )8 = 58a8

  15. Using Power Rule (b) EXAMPLE 5 Using Power Rule (b) Use power rule (b) to simplify each expression. ( a ) ( 4n)7 = 47 n7 ( b ) 2 ( x9 y4)5 = 2 ( x45 y20) = 2 x45 y20 ( c ) 3 ( 2 a3 bc4)2 = 3 ( 22a6 b2c8) = 3 ( 4 a6 b2c8) = 12 a6 b2c8

  16. The Power Rule CAUTION Power rule (b) does not apply to a sum. ( x + 3 )2≠x2 + 32 Error You will learn how to work with ( x + 3 )2 in more advanced mathematics courses.

  17. m 2 a 3 a m 32 b 4 b m 42 Power Rule (c) for Exponents Power Rule (c) for Exponents If m is a positive integer, then = (Raise a quotient to a power by raising both the numerator and the denominator to the power. The denominator cannot be 0.) Example: =

  18. 53 125 ( a ) = = 83 512 ( 3a9)2 = ( 7 b1c3)2 3a9 32 a18 9a18 2 ( b ) = = 3 7 bc3 72b2 c6 49 b2 c6 5 8 Using Power Rule (c) EXAMPLE 6 Using Power Rule (c) Simplify each expression.

  19. 2 3 32 a a m m 4 42 = = b b m Rules for Exponents Rules for Exponents If m and n are positive integers, then Product Rule am • an = am + n34 • 32 = 34 + 2 = 36 Power Rule (a) ( am )n = am n( 35 )2 = 35 • 2 = 310 Power Rule (b) ( a b )m = am bm( 5a )8 = 58 a8 Power Rule (c) (b≠ 0 ) Examples

  20. The Product Rule and Power Rules for Exponents Chapter 8 Section 1 – Completed Written by John T. Wallace

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