nth Roots & Rational Exponents

Q: Which part is the root and which is the power?

In $x^{m/n}$, the denominator $n$ is the root and the numerator $m$ is the power. Root the base, then raise to the numerator.

A fractional exponent is just a root in disguise. $\displaystyle x^{\frac{1}{n}}$ means the $n$ th root of $x$ , and $\displaystyle x^{\frac{m}{n}}$ means take the $n$ th root and raise it to the $m$ .

This bridges radicals and exponents so all the exponent rules apply to roots. Reading the fraction correctly — denominator is the root, numerator is the power — is the whole skill.

Reading a rational exponent

In $\displaystyle x^{\frac{m}{n}}$ , the denominator $n$ is the root and the numerator $m$ is the power. So $\displaystyle 8^{\frac{2}{3}}$ is the cube root of $8$ , squared.

Take the root first to keep numbers small: $\sqrt[3]{8} = 2$ , then $2^2 = 4$ . Doing the power first works too but the numbers get larger.

Why roots are exponents

Defining $\displaystyle x^{\frac{1}{n}}$ as the $n$ th root makes the exponent rules hold: $\displaystyle \left(x^{\frac{1}{n}}\right)^n = x^{\frac{n}{n}} = x$ , exactly what a root should do.

So you can add, multiply, and distribute rational exponents with the same rules you use for integer exponents.

Worked examples

Example 1: evaluating a rational exponent

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

Denominator is the root

\sqrt[3]{8} = 2

Numerator is the power

2^2

Simplify

4

Answer: $4$

Example 2: a unit fraction exponent

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

Denominator 2 means square root

\sqrt{16}

Simplify

4

Answer: $4$

Try one yourself

Common questions

Which part is the root and which is the power?

In $\displaystyle x^{\frac{m}{n}}$ , the denominator $n$ is the root and the numerator $m$ is the power. Root the base, then raise to the numerator.

Should I take the root or the power first?

Either works, but taking the root first keeps the numbers smaller and easier.

Why write a root as an exponent?

It lets you apply all the exponent rules to radicals, which simplifies expressions and equations involving roots.

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nth Roots & Rational Exponents

Reading a rational exponent

Why roots are exponents

Worked examples

Example 1: evaluating a rational exponent

Example 2: a unit fraction exponent

Try one yourself

Common questions

Keep learning

Operations with Radicals

Graphing Radical Functions

Product & Quotient Rules