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

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

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 5 4. 5 4 = 5 • 5 • 5 • 5 = 625

Evaluating Exponential Expressions EXAMPLE 2 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

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

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 4 • 3 2 = 3 4 + 2 = 3 6

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 • 3 2 ≠ 9 6 3 4 • 3 2 = 3 6

Using the Product Rule EXAMPLE 3 Using the Product Rule Use the product rule for exponents to find each product, if possible. ( a ) 6 2 • 6 7 = 6 2 + 7 = 6 9 by the product rule. ( b ) ( – 7 ) 1 ( – 7 ) 5 ( b ) ( – 7 ) 1 ( – 7 ) 5 = ( – 7 ) 1 + 5 = ( – 7 ) 6 by the product rule. ( c ) 4 7 • 3 2 The product rule doesn’t apply. The bases are different. ( d ) x 9 • x 5 = x 9 + 5 = x 14 by the product rule.

Using the Product Rule EXAMPLE 3 Using the Product Rule Use the product rule for exponents to find each product, if possible. ( e ) 8 2 + 8 3 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 15 by the product rule.

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

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

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

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 2 • 7 = 3 14 ( b ) ( 6 5 ) 9 = 6 5 • 9 = 6 45 ( c ) ( w 4 ) 2 = w 4 • 2 = w 8

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

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 ( 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

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

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

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 =

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 4 • 3 2 = 3 4 + 2 = 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 =

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