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Ch 8 Sec 1: Slide #1 Columbus State Community College Chapter 8 Section 1 The Product Rule and Power Rules for Exponents

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Ch 8 Sec 1: Slide #2 The Product Rule and Power Rules for Exponents 1.Review the use of exponents. 2.Use the product rule for exponents. 3.Use the exponent rule ( a m ) n = a m n. 4.Use the exponent rule ( a b ) m = a m b m. 5.Use the exponent rule =. a m b m a b m

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Ch 8 Sec 1: Slide #3 Review of Using Exponents EXAMPLE 1 Review of Using Exponents Write 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 = = 625

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Ch 8 Sec 1: Slide #4 Evaluating Exponential Expressions EXAMPLE 2 Evaluating Exponential Expressions Evaluate each exponential expression. Name the base and the exponent. ( a ) 2 4 = = 16 BaseExponent 24 ( b ) – 2 4 = – ( ) = – 1624 ( c ) ( – 2 ) 4 = ( – 2 )( – 2 )( – 2 )( – 2 ) = 16– 2– 24

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Ch 8 Sec 1: Slide #5 Understanding the Base It is important to understand the difference between parts (b) and (c) of Example 2. In – 2 4 the lack of parentheses shows that the exponent 4 applies only to the base 2. CAUTION ExpressionBaseExponentExample – a n ( – a ) n a – a n n – 5 2 = – ( 5 5 ) = – 25 ( – 5 ) 2 = ( – 5 ) ( – 5 ) = 25 In ( – 2 ) 4 the parentheses show that the exponent 4 applies to the base – 2. In summary, – a m and ( – a ) m mean different things. The exponent applies only to what is immediately to the left of it.

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Ch 8 Sec 1: Slide #6 Product Rule for Exponents If m and n are positive integers, then a m a n = a m + n (Keep the same base and add the exponents.) Example: = = 3 6

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Ch 8 Sec 1: Slide #7 Common Error Using the Product Rule Avoid the common error of multiplying the bases when using the product rule. Keep the same base and add the exponents. CAUTION = ≠ 9 6

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Ch 8 Sec 1: Slide #8 ( b ) ( – 7 ) 1 ( – 7 ) 5 Using the Product Rule EXAMPLE 3 Using the Product Rule Use the product rule for exponents to find each product, if possible. ( a ) = = 6 9 by the product rule. = ( – 7 ) = ( – 7 ) 6 by the product rule. ( c ) The product rule doesn’t apply. The bases are different. ( d ) x 9 x 5 = x = x 14 by the product rule.

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Ch 8 Sec 1: Slide #9 ( f ) ( 5 m n 4 ) ( – 8 m 6 n 11 ) Using the Product Rule EXAMPLE 3 Using the Product Rule Use the product rule for exponents to find each product, if possible. ( e ) The product rule doesn’t apply because this is a sum. = – 40 m 7 n 15 by the product rule. = ( 5 – 8 ) ( m m 6 ) ( n 4 n 11 )using the commutative and associative properties.

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Ch 8 Sec 1: Slide #10 Product Rule and Bases The bases must be the same before we can apply the product rule for exponents. CAUTION

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Ch 8 Sec 1: Slide #11 Understanding Differences in Exponential Expressions Be sure you understand the difference between adding and multiplying exponential expressions. Here is a comparison. Adding expressions3 x x 4 = 5 x 4 Multiplying expressions( 3 x 4 ) ( 2 x 5 ) = 6 x 9 CAUTION

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Ch 8 Sec 1: Slide #12 Power Rule (a) for Exponents If m and n are positive integers, then ( a m ) n = a m n (Raise a power to a power by multiplying exponents.) Example: ( 3 5 ) 2 = = 3 10

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Ch 8 Sec 1: Slide #13 ( b ) ( 6 5 ) 9 Using Power Rule (a) EXAMPLE 4 Using Power Rule (a) Use power rule (a) to simplify each expression. Write answers in exponential form. ( a ) ( 3 2 ) 7 = = 3 14 = = 6 45 ( c ) ( w 4 ) 2 = w 4 2 = w 8

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Ch 8 Sec 1: Slide #14 Power Rule (b) for Exponents If m is a positive integer, then ( a b ) m = a m b m (Raise a product to a power by raising each factor to the power.) Example: ( 5a ) 8 = 5 8 a 8

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Ch 8 Sec 1: Slide #15 ( b ) 2 ( x 9 y 4 ) 5 Using Power Rule (b) EXAMPLE 5 Using Power Rule (b) Use power rule (b) to simplify each expression. ( a ) ( 4n ) 7 = 4 7 n 7 = 2 ( x 45 y 20 ) = 2 x 45 y 20 ( c ) 3 ( 2 a 3 b c 4 ) 2 = 3 ( 2 2 a 6 b 2 c 8 ) = 3 ( 4 a 6 b 2 c 8 ) = 12 a 6 b 2 c 8

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Ch 8 Sec 1: Slide #16 The Power Rule Power rule (b) does not apply to a sum. ( x + 3 ) 2 ≠ x Error You will learn how to work with ( x + 3 ) 2 in more advanced mathematics courses. CAUTION

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Ch 8 Sec 1: Slide #17 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: = a m b m a b m

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Ch 8 Sec 1: Slide #18 Using Power Rule (c) EXAMPLE 6 Using Power Rule (c) Simplify each expression. ( a ) ( b ) 3a 93a 9 7 b c 3 2 = = ( 3a 9 ) 2 ( 7 b 1 c 3 ) 2 = 3 2 a b 2 c 6 = 9 a b 2 c 6 =

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Ch 8 Sec 1: Slide #19 Rules for Exponents If m and n are positive integers, then Product Rule a m a n = a m + n = = 3 6 Power Rule (a) ( a m ) n = a m n ( 3 5 ) 2 = = 3 10 Power Rule (b) ( a b ) m = a m b m ( 5a ) 8 = 5 8 a 8 Power Rule (c) ( b ≠ 0 ) a m b m a b m = Examples =

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

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