Beginning Algebra, 11th Edition

OBJECTIVE 3 Use the rule Consider the following.

Product rule for exponents

The product of the exponents in , , gives the exponent in Also

Definition of exponent

Product rule

Add the exponents.

and 2 # 4 = 8.These examples suggest power rule (a) for exponents.

= 58 ,

= 52 +^2 +^2 +^2

15224 = 52 # 52 # 52 # 52

183223 # 2 86.

18322 = 18321832 = 83 +^3 = 86

1 am 2 namn.

298 CHAPTER 5 Exponents and Polynomials

Power Rule (a) for Exponents

For any positive integers mand n,

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

Example: 13224 = 32

# 4

= 38

1 am 2 namn.

Using Power Rule (a)

Use power rule (a) for exponents to simplify.

(a) (b) (c)

NOW TRY

OBJECTIVE 4 Use the rule Consider the following.

Definition of exponent

Commutative and associative properties

Definition of exponent

This example suggests power rule (b) for exponents.

= 43 #x^3

= 14 # 4 # 421 x#x#x 2

14 x 23 = 14 x 214 x 214 x 2

1 ab 2 mambm.

12523 = 25 #^3 = 215 15722 = 57122 = 514 1 x^225 = x^2152 = x^10

NOW TRY EXAMPLE 4

EXERCISE 4
Simplify.

(a) 14725 (b) 1 y^427

NOW TRY ANSWERS

4. (a) 435 (b)y^28

Power Rule (b) for Exponents

For any positive integer m,

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

Example: 12 p 25 = 25 p^5

1 ab 2 mambm.

Using Power Rule (b)

Use power rule (b) for exponents to simplify.

(a)

Power rule (b)

= 9 x^2 y^232 = 3 # 3 = 9

= 32 x^2 y^2

13 xy 22

EXAMPLE 5

(b)

Power rule (b)

= 5 p^2 q^2 Multiply.

= 51 p^2 q^22

51 pq 22

(c)

Power rule (b)

Power rule (a)

= 48 m^8 p^123 # 24 = 3 # 16 = 48

= 3 # 24 m^8 p^12

= 33241 m^2241 p^3244

312 m^2 p^324

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