Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide 2 Exponents and Polynomials Chapter 5

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Objectives 1.Use exponents. 2.Use the product rule for exponents. 3.Use the rule (a m ) n = a mn. 4.Use the rule (ab) m = a m b m. 5.Use the rule (a/b) m = a m /b m. 6.Use combinations of the rules for exponents. 7.Use the rules for exponents in a geometry application.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using Exponents Exponent (or Power) Base 6 factors of 2 2 = 2 · 2 · 2 · 2 · 2 · 2 6 The exponential expression is 2 6, read “2 to the sixth power” or simply “2 to the sixth.” Example 1 Write 2 · 2 · 2 · 2 · 2 · 2 in exponential form and evaluate. Since 2 occurs as a factor 6 times, the base is 2 and the exponent is 6. = 64

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using Exponents (a) 2 = 2 · 2 · 2 · 2 4 = 16 BaseExponent 24 (b) – 2 4 = – = – 1 · 2 = – 1 · 2 · 2 · 2 · 2 (c) (– 2) 4 = 16 – 24 = (– 2) (– 2) (– 2) (– 2) Example 2 Evaluate. Name the base and the exponent.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide 7 ExpressionBaseExponentExample 5.2 The Product Rule and Power Rules for Exponents Using Exponents CAUTION – a n (– a) n In summary, and are not necessarily the same. – a n (– a) n a n n – 5 2 (– 5) 2 = – ( 5 · 5 ) = – 25 = (– 5) (– 5) = 25

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Product Rule for Exponents Product Rule for Exponents For any positive integers m and n, a m · a n = a m + n (Keep the same base and add the exponents.) 5 3 · 5 4 Example: ( 5 · 5 · 5 ) = ( 5 · 5 · 5 · 5 ) = 5. 7

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Product Rule for Exponents CAUTION Do not multiply the bases when using the product rule. Keep the same base and add the exponents. Example:

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Product Rule for Exponents = (a)7 · 7 = 7 13 = y (c)y · y = y 6 5 = y · y 1 = (– 2) (b)(– 2) (– 2)= (– 2) 9 Example 3 Use the product rule for exponents to simplify, if possible.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Product Rule for Exponents = n (d)n · n= n 6 The product rule does not apply because the bases are different. 22 (e)3 · 2 The product rule does not apply because it is a sum, not a product. 32 (f)2 + 2 = = 12 = 9 · 4 = 36 Example 3 (concluded) Use the product rule for exponents to simplify, if possible.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Product Rule for Exponents CAUTION The bases of the factors must be the same before we can apply the product rule for exponents.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Product Rule of Exponents Add the exponents. Example 4 Multiply; product rule Commutative and associative properties

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Product Rule for Exponents CAUTION Be sure you understand the difference between adding and multiplying exponential expressions. For example, 22 7k + 3k 22 = ( )k = 10k, = ( 7 · 3 )k = 21k. 22 7k 3k but, Add. Multiply.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Rule (a m ) n = a mn Power Rule (a) for Exponents For any positive integers m and n, mn ( a ) Example: 32 ( 4 ) 3 · 2 = 4 (Raise a power to a power by multiplying exponents.) 6 = 4. m n = a.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Rule (a m ) n = a mn Example 5 Use power rule (a) for exponents to simplify. (a) ( 3 ) 25 = 3 2 · 5 = 3 10 (b) ( 4 ) 86 = 4 8 · 6 = 4 48 (c) ( n ) 73 = n 7 · 3 = n 21 (d) ( 2 ) 58 = 2 5 · 8 = 2. 40

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Rule (ab) m = a m b m Power Rule (b) for Exponents For any positive integer m, m ( ab ) m = a b. Example: 3 ( 5h ) (Raise a product to a power by raising each factor to the power.) 3 = 5 h. 3

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Rule (ab) m = a m b m Example 6 Use power rule (b) for exponents to simplify. (a) ( 2abc ) 4 = 2 a b c 4444 = 16 a b c 444 Power rule (b) = 5 ( x y ) 26 Power rule (b) (b) 5 ( x y ) 23 = 5x y 26

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Rule (ab) m = a m b m Example 6 (continued) Use power rule (b) for exponents to simplify. Power rule (b) (c) 7 ( 2m n p ) 7 53 = 7 [ 2 ( m ) ( n ) ( p ) ] Power rule (a) = 7 [ 8 m n p ] = 56 m n p

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Rule (ab) m = a m b m Example 6 (concluded) Use power rule (b) for exponents to simplify. Power rule (b) (d) ( – 3 ) 5 4 Power rule (a) = ( – 1 · 3 ) 5 4 = ( – 1 ) ( 3 ) 5 45 = – 1 · 3 20 = – 3 20 – a = – 1 · a

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Rule (ab) m = a m b m CAUTION Power rule (b) does not apply to a sum:

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Rule (a/b) m = a m /b m Power Rule (c) for Exponents For any positive integer m, (Raise a quotient to a power by raising both the numerator and the denominator to the power.) m a b m a b m = Example: = ( b ≠ 0 ).

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Rule (a/b) m = a m /b m Example 7 Use power rule (c) for exponents to simplify = (a) = 7 x y 7 x y 7 = (b) y ≠ 0

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Rules for Exponents For any positive integers m and n:Examples Product rule a m · a n = a m + n 3 4 · 3 5 = 3 9 Power rules mn ( a ) m n = a (a) 45 ( 2 ) 20 = 2 m m (b) ( ab ) = a b 3 ( 4k ) 3 = 4 k 3 (c) m a b m a b m = ( b ≠ 0 ) =

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using Combinations of the Rules for Exponents Example 8 Simplify each expression. Power rule (c) Multiply fractions. (a) · = · = 2 3 · 4 · Product rule =

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using Combinations of the Rules for Exponents Example 8 (continued) Simplify each expression. Product rule (b) ( 7m n ) ( 7mn ) 2235 = ( 7m n ) 28 Power rule (b) = 7 m n 1688

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using Combinations of the Rules for Exponents Example 8 (continued) Simplify each expression. Power rule (b) (c) ( 2x y ) ( 2 x y ) = ( 2 ) ( x ) ( y ) · ( 2 ) ( x ) ( y ) = 2 · x · y · 2 · x · y Power rule (a) = 2 · 2 · x · x · y · y Commutative and associative properties = 2 · x · y Product rule

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using Combinations of the Rules for Exponents Example 8 (concluded) Simplify each expression. Power rule (b) (d) ( – g h ) = ( – 1 g h ) = ( – 1 ) ( g ) ( h ) Product rule = ( – 1 ) ( g ) ( h ) = – 1 g h 12 17

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Rules for Exponents in a Geometry Application Example 9 Find a polynomial that represents the area of the geometric figure. (a)Use the formula for the area of a rectangle, A = LW. A = ( 4x )( 2x ) 32 A = 8x 5 Product rule 4x4x 3 2x2x 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. 5.2 – Slide The Product Rule and Power Rules for Exponents Using the Rules for Exponents in a Geometry Application Example 9 Find a polynomial that represents the area of the geometric figure. A = 12n 9 Product rule 8n8n 5 3n 4 A = ( 3n ) ( 8n ) (b)Use the formula for the area of a triangle, A = LW. 1 2 A = ( 24n ) 91 2