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**Columbus State Community College**

Chapter 8 Section 1 The Product Rule and Power Rules for Exponents

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**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 = a b m a m b m

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**Review of Using Exponents**

EXAMPLE 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 5 4. 5 4 = 5 • 5 • 5 • 5 = 625

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**Evaluating Exponential Expressions**

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

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**Understanding the Base**

CAUTION 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. 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. Expression Base Exponent Example – a n a n – 5 2 = – ( 5 • 5 ) = – 25 ( – a ) n – a n ( – 5 ) 2 = ( – 5 ) ( – 5 ) = 25

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**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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**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. 3 4 • ≠ 9 6 3 4 • = 3 6

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

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Using the Product Rule EXAMPLE 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. ( f ) ( 5 m n 4 ) ( – 8 m 6 n 11 ) = ( 5 • – 8 ) • ( m m 6 ) • ( n 4 n 11 ) using the commutative and associative properties. = – 40 m 7 n by the product rule.

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

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**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 x x 4 = 5 x 4 Multiplying expressions ( 3 x 4 ) ( 2 x 5 ) = 6 x 9

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**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 5 • 2 = 3 10

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Using Power Rule (a) EXAMPLE Using Power Rule (a) Use power rule (a) to simplify each expression. Write answers in exponential form. ( a ) ( 3 2 ) 7 = • 7 = ( b ) ( 6 5 ) 9 = • 9 = ( c ) ( w 4 ) 2 = w 4 • 2 = w 8

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**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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Using Power Rule (b) EXAMPLE Using Power Rule (b) Use power rule (b) to simplify each expression. ( a ) ( 4n ) 7 = n 7 ( b ) 2 ( x 9 y 4 ) 5 = 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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The Power Rule CAUTION Power rule (b) does not apply to a sum. ( x ) 2 ≠ x Error You will learn how to work with ( x ) 2 in more advanced mathematics courses.

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**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 b m a m b m 3 4 2 3 2 4 2

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**3 2 Using Power Rule (c) EXAMPLE 6 Using Power Rule (c)**

Simplify each expression. ( a ) 5 8 3 = 5 3 8 3 = 125 512 ( b ) 3a 9 7 b c 3 2 ( 3a 9 ) 2 ( 7 b 1 c 3 ) 2 = 3 2 a 18 7 2 b 2 c 6 = 9 a 18 49 b 2 c 6 =

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**m 2 Rules for Exponents 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 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 ) Examples a m b m a b m = 3 2 4 2 3 4 2 =

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**The Product Rule and Power Rules for Exponents**

Chapter 8 Section 1 – Completed Written by John T. Wallace

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