9.8 Rational Exponents

Learning Objectives

By the end of this section, you will be able to:

• Simplify expressions with a^1/n
• Simplify expressions with a^m/n
• Use the Laws of Exponents to simply expressions with rational exponents

Be Prepared 9.21

Before you get started, take this readiness quiz.

Add: 7/15+5/12.
If you missed this problem, review Example 1.81.

Be Prepared 9.22

Simplify: (4x²y⁵)³.
If you missed this problem, review Example 6.24.

Be Prepared 9.23

Simplify: 5^−3.
If you missed this problem, review Example 6.89.

Simplify Expressions with a^1/n

Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.

The Power Property for Exponents says that (a^m)ⁿ=a^m⋅n when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number p such that (8p)3=8. We will use the Power Property of Exponents to find the value of p.

But we know also (³√8)³=8. Then it must be that 8^1/3= ³ √8.

This same logic can be used for any positive integer exponent n to show that a^1/n=ⁿ√a

Rational Exponent a^1/n

If ⁿ√a is a real number and n≥2, a^1/n=ⁿ√a.

There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.

In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.

Be careful of the placement of the negative signs in the next example. We will need to use the property a⁻ⁿ=1 /aⁿ in one case.

Simplify Expressions with a^m/n

Let’s work with the Power Property for Exponents some more.

Suppose we raise a^1/n to the power m.

Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller.

Remember that b- ^p =1/b^p. The negative sign in the exponent does not change the sign of the expression.

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

The same laws of exponents that we already used apply to rational exponents, too. We will list the Exponent Properties here to have them for reference as we simplify expressions.

When we multiply the same base, we add the exponents.

We will use the Power Property in the next example.

The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

Sometimes we need to use more than one property. In the next two examples, we will use both the Product to a Power Property and then the Power Property.

We will use both the Product and Quotient Properties in the next example.

Section 9.8 Exercises

Practice Makes Perfect

Simplify Expressions with a^1/n

In the following exercises, write as a radical expression.

In the following exercises, simplify.

Simplify Expressions with a^m/n

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Everyday Math

600.Landscaping Joe wants to have a square garden plot in his backyard. He has enough compost to cover an area of 144 square feet. Simplify 144^1/2 to find the length of each side of his garden.

601.Landscaping Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify 24212 to find the length of each side of his patio.Round to the nearest tenth of a foot.

602.Gravity While putting up holiday decorations, Bob dropped a decoration from the top of a tree that is 12 feet tall. Simplify 12^1/2/16 ^1/2 to find how many seconds it took for the decoration to reach the ground. Round to the nearest tenth of a second.

603.Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Simplify 1024121612 to find how many seconds it took for the flare to reach the water.

Writing Exercises

604.Show two different algebraic methods to simplify 4^3/2. Explain all your steps.

605.Explain why the expression (−16) ^3/2 cannot be evaluated.