Intermediate Algebra (11th edition)

(d)

Power rule

Power rule

Quotient rule

= Definition of negative exponent

y38/9

x^4

= x-^4 y38/9 4 - A- 92 B=^369 + 92 =^389

= x-8/3-4/3y^4 -^1 - 2/9^2

=

x-8/3y^4

x4/3y-2/9

=

1 x^42 - 2/3 1 y-^62 - 2/3

1 x-^22 - 2/3 1 y1/3 2 - 2/3

a

x^4 y-^6

x-^2 y1/3

b

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440 CHAPTER 8 Roots, Radicals, and Root Functions

The same result is obtained if we simplify within the parentheses first.

Quotient rule

Power rule

Power rule

Definition of negative exponent

(e)

Distributive property

Product rule

Add exponents.

Lowest terms in exponents

NOW TRY

=m^2 - m

=m8/4- m4/4

=m3/4+5/4-m3/4+1/4

=m3/4 1 m5/4 2 - m3/4 1 m1/4 2

m3/4 1 m5/4- m1/4 2

=

y38/9

x^4

=x-^4 y38/9

= 1 x^62 - 2/3 1 y-19/3 2 - 2/3

= 1 x^6 y-19/3 2 - 2/3 - 6 - 31 =-^183 -^13 =-^193

= 1 x^4 -^1 -^22 y-^6 - 1/3 2 - 2/3

a

x^4 y-^6

x-^2 y1/3

b

• 2/3

Use parentheses

to avoid errors.

CAUTION Use the rules of exponents in problems like those in Example 5.Do

not convert the expressions to radical form.

Do not make the common

mistake of multiplying

exponents in the first step.

Applying Rules for Rational Exponents

Write all radicals as exponentials, and then apply the rules for rational exponents. Leave

answers in exponential form. Assume that all variables represent positive real numbers.

(a)

Convert to rational exponents.

Product rule

Write exponents with a common denominator.

= x11/12 Add exponents.

= x8/12+3/12

= x2/3+1/4

= x2/3 #x1/4

23 x^2 # 24 x

EXAMPLE 6

NOW TRY
EXERCISE 5
Write with only positive
exponents. Assume that all
variables represent positive
real numbers.

(a) (b)

(c)

(d)

(e)y2/3 1 y1/3+y5/3 2

a

2 x1/2y-2/3

x-3/5y-1/5

b

• 3

1 r2/3t1/4 28

t

9 3/5

9 7/5

5 1/4# 5 2/3

NOW TRY ANSWERS

5. (a) (b)

(c) (d)

(e)y+y7/3

y7/5

8 x33/10

r16/3t

1

9 4/5

5 11/12