Dealing with fractional exponents

How fractional exponents are related to roots

In this lesson we’ll work with both positive and negative fractional exponents. Remember that when ???a??? is a positive real number, both of these equations are true:

???x^{-a}=\frac{1}{x^a}???

???\frac{1}{x^{-a}} = x^a???

The rule for fractional exponents:

When you have a fractional exponent, the numerator is the power and the denominator is the root. In the variable example ???x^{\frac{a}{b}}???, where ???a??? and ???b??? are positive real numbers and ???x??? is a real number, ???a??? is the power and ???b??? is the root.

???x^{\frac{a}{b}}??? ???=??? ???\sqrt[b]{x^a}???

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An example where we change the fractional exponent into a root in order to simplify the expression

Example

Simplify the expression.

???4^{\frac{3}{2}}???

In the fractional exponent, ???3??? is the power and ???2??? is the root, which means we can rewrite the expression as

???\sqrt{4^3}???

???\sqrt{4 \cdot 4 \cdot 4}???

???\sqrt{64}???

???8???

in a fractional exponent, think of the numerator as an exponent, and the denominator as the root

Another rule for fractional exponents:

To make a problem easier to solve you can break up the exponents by rewriting them. For example, you can write ???x^{\frac{a}{b}}??? as

???\left[(x)^a\right]^{\frac{1}{b}}???

or as

???\left[(x)^{\frac{1}{b}}\right]^a???

Let’s do a few examples.

Let's do another example.

Example

Simplify the expression.

???\left(\frac{1}{9}\right)^{\frac{3}{2}}???

???9??? is a perfect square so it can simplify the problem to find the square root first. We can rewrite the expression by breaking up the exponent.

???\left[\left(\frac{1}{9}\right)^{\frac{1}{2}}\right]^3???

Raising a value to the power ???1/2??? is the same as taking the square root of that value, so we get

???\left[\sqrt{\frac{1}{9}}\right]^3???

???\left(\frac{\sqrt{1}}{\sqrt{9}}\right)^3???

???\left(\frac{1}{3}\right)^3???

This is the same as

???\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)???

???\frac{1}{27}???

How to get rid of fractional exponents

Example

Write the expression without fractional exponents.

???\left(\frac{1}{6}\right)^{\frac{3}{2}}???

We can rewrite the expression by breaking up the exponent.

???\left[\left(\frac{1}{6}\right)^3\right]^{\frac{1}{2}}???

???\left(\frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6}\right)^{\frac{1}{2}}???

Raising a value to the power ???1/2??? is the same as taking the square root of that value, so we get

???\sqrt{\frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6}}???

???\sqrt{\frac{1}{216}}???

???\frac{\sqrt{1}}{\sqrt{216}}???

???\frac{1}{\sqrt{36 \cdot 6}}???

???\frac{1}{\sqrt{36} \sqrt{6}}???

???\frac{1}{6\sqrt{6}}???

We need to rationalize the denominator.

???\frac{1}{6\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}???

???\frac{\sqrt{6}}{6 \cdot 6}???

???\frac{\sqrt{6}}{36}???

What happens if you have a negative fractional exponent?

You should deal with the negative sign first, then use the rule for the fractional exponent.

Example

Write the expression without fractional exponents.

???4^{-\frac{2}{5}}???

First, we’ll deal with the negative exponent. Remember that when ???a??? is a positive real number, both of these equations are true:

???x^{-a}=\frac{1}{x^a}???

???\frac{1}{x^{-a}} = x^a???

Therefore,

???4^{-\frac{2}{5}}???

???\frac{1}{4^{\frac{2}{5}}}???

In the fractional exponent, ???2??? is the power and ???5??? is the root, which means we can rewrite the expression as

???\frac{1}{\sqrt[5]{4^2}}???

???\frac{1}{\sqrt[5]{16}}???