Chapter Thirteen — Radical Expressions and Equations

The Humongous Book of Algebra Problems

281

Rational Exponents

Fractional powers are radicals in disguise

13.13 Write x

a/b

as a radical expression. Assume that a and b are natural numbers.

Rational exponents (that is, fractional exponents) indicate radical expressions.

The denominator of the rational power (b in this problem) represents the

index of the radical, and the numerator of the power represents either the

exponential power of the radicand or the power to which the entire expression

is raised: and .

13.14 Write 4

3/2

as a radical expression and simplify it.

According to Problem 13.13, . In this problem, x = 4, a = 3, and

b = 2.

13.15 Write 250

1/3

as a radical expression and simplify it.

According to Problem 13.13, . In this problem, x = 250, a = 1, and

b = 3.

Notice that 125 is a perfect cube and a factor of 250.

13.16 Write as an exponential expression with base 7.

The radicand 49 is equal to 7

2

, so . According to Problem 13.13,

. In this problem, x = 7, a = 2, and b = 5.

You can use

either formula

to rewrite x

a/b

.

You’ll probably use

more

often because x

a

tends

to be a large number

that’s hard to

simplify.

Chapter Thirteen — Radical Expressions and Equations

The Humongous Book of Algebra Problems

282

13.17 According to Problem 13.1, . Verify the statement using rational

exponents.

Begin by writing as an exponential expression with base 4.

Rewrite the exponential expression using a rational exponent.

13.18 Write (12x

3

y

2

)

1/2

as a radical expression and simplify.

The entire quantity 12x

3

y

2

is raised to the rational exponent , so construct a

radical expression with radicand 12x

3

y

2

and index 2.

Simplify the radical expression by identifying perfect square factors.

13.19 Write (9x)

1/2

y

7/4

as a radical expression and simplify it.

Note that 9 and x are both raised to the power.

(9x)

1/2

y

7/4

= 9

1/2

x

1/2

y

7/4

Rewrite the expression as a product of radical expressions.

Do this the

same way Problem

13.16 turned

into an exponential

expression with

base 7.

Start Free Trial

No credit card required