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


Exponents and Polynomials

Chapter 5


5.2

The Product Rule and Power Rules

for Exponents


5.2 The Product Rule and Power Rules for Exponents

Objectives

1. Use exponents.2. Use the product rule for exponents.3. Use the rule (am)n = amn. 4. Use the rule (ab)m = ambm.5. Use the rule (a/b)m = am/bm.6. Use combinations of the rules for exponents.7. Use the rules for exponents in a geometry application.


5.2 The Product Rule and Power Rules for ExponentsUsing Exponents

Exponent (or Power)

Base 6 factors of 2

2 = 2 · 2 · 2 · 2 · 2 · 2 6

The exponential expression is 26, 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



(a) 2 = 2 · 2 · 2 · 24 = 16

Base Exponent

2 4

(b) – 2 4

= – 16

2 44= – 1 · 2 = – 1 · 2 · 2 · 2 · 2

(c) (– 2) 4

= 16– 2 4

= (– 2) (– 2) (– 2) (– 2)

Example 2 Evaluate. Name the base and the exponent.


Expression Base Exponent Example


CAUTION

– a n

(– a) n

In summary, and are not necessarily the same.– a n (– a) n

a n

(– a) n

– 5 2

(– 5) 2

= – ( 5 · 5 ) = – 25

= (– 5) (– 5) = 25



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.)

53 · 54

Example:( 5 · 5 · 5 ) = 5 3 + 4( 5 · 5 · 5 · 5 ) = 5 .7




CAUTION

Do not multiply the bases when using the product rule. Keep the same base and add the exponents.

Example: 3 4 7 75 5 5 not 25 .




= 7 5 + 85 8(a) 7 · 7 = 7 13

= y 1 + 55(c) y · y = y 65= y · y1

= (– 2) 4 + 54 5(b) (– 2) (– 2) = (– 2) 9

Example 3 Use the product rule for exponents to simplify, if possible.




= n 2 + 42 4(d) n · n = n 6

The product rule does not apply because the bases are different.

2 2(e) 3 · 2

The product rule does not apply because it is a sum, not a product.

3 2(f) 2 + 2

= 8 + 4 = 12

= 9 · 4 = 36

Example 3 (concluded) Use the product rule for exponents to simplify, if possible.




CAUTION

The bases of the factors must be the same before we can apply the product rule for exponents.



Using the Product Rule of Exponents

Add the exponents.

3 6Multiply 5 4 .y y

Example 4

3 65 4y y

3 65 4 y y

3 620y

Multiply; product rule

Commutative and associative properties

920y




CAUTION

Be sure you understand the difference between adding and multiplying exponential expressions. For example,

2 27k + 3k 22= ( 7 + 3 )k = 10k ,

42 + 2= ( 7 · 3 )k = 21k .2 27k 3kbut,

Add.

Multiply.



Using the Rule (am)n = amn

Power Rule (a) for ExponentsFor any positive integers m and n,

m n( a )

Example: 3 2( 4 ) 3 · 2= 4

(Raise a power to a power by multiplying exponents.)

6= 4 .

m n= a .



Using the Rule (am)n = amn

Example 5

Use power rule (a) for exponents to simplify.

(a) ( 3 )2 5 = 3 2 · 5 = 3 10

(b) ( 4 )8 6 = 4 8 · 6 = 4 48

(c) ( n )7 3 = n 7 · 3 = n 21

(d) ( 2 )5 8 = 2 5 · 8 = 2.40



Using the Rule (ab)m = ambm

Power Rule (b) for ExponentsFor any positive integer m,

m( ab ) m m= a b .

Example: 3( 5h )

(Raise a product to a power by raising each factor to the power.)

3 = 5 h .

3




Example 6Use power rule (b) for exponents to simplify.

(a) ( 2abc ) 4 = 2 a b c4 4 4 4

= 16 a b c 4 4 4

Power rule (b)

= 5 ( x y )2 6 Power rule (b)(b) 5 ( x y ) 23

= 5x y 2 6




Example 6 (continued)Use power rule (b) for exponents to simplify.

Power rule (b)

(c) 7 ( 2m n p )75 3

= 7 [ 2 ( m ) ( n ) ( p ) ]3 5 3 71 3 3

Power rule (a)= 7 [ 8 m n p ]3 15 21

= 56 m n p 3 15 21




Example 6 (concluded)Use power rule (b) for exponents to simplify.

Power rule (b)

(d) ( – 3 ) 54

Power rule (a)

= ( – 1 · 3 ) 54

= ( – 1 ) ( 3 ) 545

= – 1 · 3 20

= – 3 20

– a = – 1 · a




CAUTIONPower rule (b) does not apply to a sum:

2 22 2 2 25 5 , but 5 5 .y y y y



Using the Rule (a/b)m = am/bm

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.)

mab

mabm=

Example: .23

4

234 2=

( b ≠ 0 ).



Using the Rule (a/b)m = am/bm

Example 7Use power rule (c) for exponents to simplify.

425

425 4=(a)

16625

=

7xy

7xy 7=(b) y ≠ 0



Rules for ExponentsFor any positive integers m and n: Examples

Product rule am · an = am + n 34 · 35 = 39

Power rules m n( a ) m n= a (a) 4 5( 2 )20= 2

m m m(b) ( ab ) = a b 3( 4k ) 3= 4 k 3

(c)ma

b

mabm=

( b ≠ 0 ).

347

=34

7 3



Using Combinations of the Rules for Exponents

Example 8 Simplify each expression.

Power rule (c)

Multiply fractions.

(a) · 3523

4= ·

2342

53 1

=23 ·

4 ·2

53 1

Product rule= 342

7




Example 8 (continued) Simplify each expression.

Product rule

(b) ( 7m n ) ( 7mn )22 3 5

= ( 7m n )2 8

Power rule (b)= 7 m n 16 88




Example 8 (continued) Simplify each expression.

Power rule (b)

(c) ( 2x y ) ( 2 x y ) 1063 7 4

= ( 2 ) ( x ) ( y ) · ( 2 ) ( x ) ( y )10 7 46 3 6 6 10 10

= 2 · x · y · 2 · x · y 10 70 406 18 6 Power rule (a)

= 2 · 2 · x · x · y · y70 6 406 10 18 Commutative and associative properties

= 2 · x · y16 88 46 Product rule




Example 8 (concluded) Simplify each expression.

Power rule (b)

(d) ( – g h ) ( – g h )3 2 5 32

= ( – 1 g h ) ( – 1 g h )3 2 5 32

= ( – 1 ) ( g ) ( h ) ( – 1 ) ( g ) ( h )2 26 3 6 15

Product rule= ( – 1 ) ( g ) ( h )5 12 17

= – 1 g h 12 17



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 )

3 2

A = 8x 5 Product rule

4x 3

2x 2



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

8n 5

3n 4

A = ( 3n ) ( 8n ) 4 512

(b) Use the formula for the area of a triangle, A = LW.12

A = ( 24n )912

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

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