Power of a Fraction and Rational Exponents

Q: Which part of a fractional exponent is the root?

The bottom. In $x^{\frac{a}{b}}$, the $b$ is the root and the $a$ is the power: $\sqrt[b]{x^{a}}$. Remember it as the root sitting at the root of the fraction.

This lesson covers two exponent ideas that finish out the rule set. First, raising a fraction to a power: the exponent applies to the top and the bottom, so $\left(\dfrac{2}{3}\right)^{2} = \dfrac{4}{9}$ . Second, fractional (rational) exponents: an exponent like $\dfrac{1}{2}$ or $\dfrac{2}{3}$ is a root in disguise.

Both rules follow from what you already know. A fraction to a power is just power of a product applied to a quotient, and rational exponents are what you get when the exponent rules are pushed to work for roots.

Power of a fraction: hit the top and the bottom

In symbols, $\left(\dfrac{x}{y}\right)^{a} = \dfrac{x^{a}}{y^{a}}$ . Raise the numerator to the power and the denominator to the power.

If the top and bottom already have exponents, the power rule takes over from there: $\left(\dfrac{x^{3}}{y^{2}}\right)^{4} = \dfrac{x^{3 \cdot 4}}{y^{2 \cdot 4}} = \dfrac{x^{12}}{y^{8}}$ . The most common mistake is applying the outside exponent to the top only.

Rational exponents are roots

In a fractional exponent, the bottom number is the root and the top number is the power: $\displaystyle x^{\frac{a}{b}} = \sqrt[b]{x^{a}}$ .

So $\displaystyle x^{\frac{1}{2}}$ is the square root of $x$ , and $\displaystyle x^{\frac{1}{3}}$ is the cube root. A quick check that this makes sense: by the power rule, $\displaystyle \left(x^{\frac{1}{2}}\right)^{2} = x^{1} = x$ — squaring it gives back $x$ , which is exactly what a square root does.

When the top number isn't $1$ , do the root first, then the power. For $\displaystyle 8^{\frac{2}{3}}$ : the cube root of $8$ is $2$ , and $2^{2} = 4$ . Root first keeps the numbers small.

Worked examples

Example 1: power of a fraction with variables

Simplify $\left(\dfrac{x^{3}}{y^{2}}\right)^{4}$ .

Apply the exponent to the top and the bottom

\dfrac{(x^{3})^{4}}{(y^{2})^{4}}

Multiply the exponents on

x

3 \cdot 4 = 12

Multiply the exponents on

y

2 \cdot 4 = 8

Put it together

\dfrac{x^{12}}{y^{8}}

Answer: $\dfrac{x^{12}}{y^{8}}$

Example 2: an exponent of one half

Evaluate $\displaystyle 16^{\frac{1}{2}}$ .

The bottom of the fraction is the root — here, a square root

\displaystyle 16^{\frac{1}{2}} = \sqrt{16}

Take the square root

4

Answer: $4$

Example 3: root first, then power

Evaluate $\displaystyle 8^{\frac{2}{3}}$ .

Bottom is the root, top is the power

\displaystyle 8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^{2}

Take the cube root of

8

\sqrt[3]{8} = 2

Square it

2^{2} = 4

Answer: $4$

Try one yourself

Common questions

Which part of a fractional exponent is the root?

The bottom. In $\displaystyle x^{\frac{a}{b}}$ , the $b$ is the root and the $a$ is the power: $\sqrt[b]{x^{a}}$ . Remember it as the root sitting at the root of the fraction.

For something like

\displaystyle 8^{\frac{2}{3}}

, do I do the root or the power first?

Either order gives the same answer, but the root first keeps the numbers small: cube root of $8$ is $2$ , then $2^{2} = 4$ . Power first means cubing later — $8^{2} = 64$ , then the cube root of $64$ is still $4$ , just with bigger arithmetic.

Does the exponent on a fraction apply to the denominator too?

Yes — top and bottom both. $\left(\dfrac{3}{4}\right)^{2} = \dfrac{9}{16}$ , not $\dfrac{9}{4}$ . Raising only the numerator is the classic error here.

How do I rewrite a radical as a fractional exponent?

The index of the root becomes the bottom of the exponent and the power stays on top: $\displaystyle \sqrt[4]{x^{3}} = x^{\frac{3}{4}}$ . A plain square root has index $2$ , so $\displaystyle \sqrt{x} = x^{\frac{1}{2}}$ .

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Power of a Fraction and Rational Exponents

Power of a fraction: hit the top and the bottom

Rational exponents are roots

Worked examples

Example 1: power of a fraction with variables

Example 2: an exponent of one half

Example 3: root first, then power

Try one yourself

Common questions

Keep learning

Power Rule

Negative Exponent Rule

Operations with Radicals

Exponential Equations