Rational Exponents and Roots

You already know that $a^{2}$ means $a \times a$ and $a^{3}$ means $a \times a \times a$ . But what does it mean to raise a number to a fractional power like $a^{\frac{1}{2}}$ or $a^{\frac{2}{3}}$ ? Rational exponents connect the world of powers and roots.

Rational Exponent

For any positive real number $a$ and rational number $\frac{m}{n}$ where $m$ and $n$ are integers and $n \neq = 0$ :
$a^{\frac{m}{n}} = n a^{m}$
The denominator of the fraction becomes the root, and the numerator becomes the power.

A helpful way to remember this: think "power over root" — the top of the fraction is the power, the bottom is the root.

Example

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

Step 1: Identify the power ( $m = 2$ ) and the root ( $n = 3$ ).

Step 2: Apply the definition:
$8^{\frac{2}{3}} = 3 8^{2} = 364 = 4$

Alternatively, you can take the root first, then apply the power — often easier with large numbers:
$8^{\frac{2}{3}} = (38)^{2} = 2^{2} = 4$

Both approaches give the same answer.

Note

When $n$ is even, $a^{\frac{1}{n}}$ refers specifically to the positive $n$ th root of $a$ . For example, $9^{\frac{1}{2}} = 3$ , not $\pm 3$ . This convention keeps exponential expressions well-defined for positive real bases.

Exam Tip

When evaluating $a^{\frac{m}{n}}$ by hand, it is usually easier to take the root first, then raise to the power. This keeps intermediate numbers smaller and more manageable.

Free preview

SL 1.7—Laws of exponents and logs

Rational Exponents and Roots

9 more sections in this topic