Fractional Exponents: Roots Written as Powers

Q: Is x to the power 1/2 the same as the square root of x?

Yes, for x ≥ 0. x^1/2 = √x because squaring it gives x^1 = x.

Why $9^{1/2}$ has to be the square root of $9$ if the exponent laws are to keep working, how to evaluate powers like $8^{2/3}$ quickly, and how negative and fractional exponents combine.

2 min read Supports 3 lessons

What could $9^{1/2}$ mean? You cannot multiply $9$ by itself half a time. But if the exponent laws are to keep working, there is only one sensible answer.

$\left(9^{1/2}\right)^2$
Square the mystery number.

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Fractional exponents

a^{1/n} = \sqrt[n]{a} \qquad a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}

The denominator is the root; the numerator is the power.

25^{1/2} = 5 \qquad 8^{1/3} = 2 \qquad 16^{1/4} = 2

Root first, then power

$a^{m/n}$ can be worked out in either order, but taking the root first keeps the numbers small.

Worked example

Evaluate $27^{2/3}$ .

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Cube root first.
$\sqrt[3]{27} = 3$

Tip

$27^{2/3}$ the other way round is $\sqrt[3]{729}$ , which is much harder to spot. Root first, power second.

Adding a negative sign

A negative fractional exponent does all three things: the minus sign means reciprocal, the denominator means root, the numerator means power.

$16^{-3/4}$
Three instructions in one exponent.

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Why write roots as powers?

Because then every exponent law applies to roots too. Simplifying $\sqrt{x} \cdot \sqrt[3]{x}$ looks awkward, but as powers it is just adding fractions:

x^{1/2} \cdot x^{1/3} = x^{\frac12 + \frac13} = x^{5/6}

Key idea

In calculus, writing $\sqrt{x}$ as $x^{1/2}$ lets the same rule for powers handle roots, so you rarely need a separate method for them.

Watch out

$a^{1/2}$ is not $\frac{a}{2}$ . $16^{1/2} = 4$ , while $\frac{16}{2} = 8$ .

Watch out

Even roots of negative numbers are not real: $(-16)^{1/2}$ has no real value. Odd roots are fine: $(-8)^{1/3} = -2$ .

Common questions

What does a fractional exponent mean?: The denominator gives a root and the numerator gives a power. So $a^{3/2}$ means the square root of $a$ , cubed.
Is x to the power 1/2 the same as the square root of x?: Yes, for $x \ge 0$ . $x^{1/2} = \sqrt{x}$ because squaring it gives $x^1 = x$ .

Practise fractional exponents

The Fractional Exponents lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What does a fractional exponent mean?

The denominator gives a root and the numerator gives a power. So $a^{3/2}$ means the square root of $a$ , cubed.

Is x to the power 1/2 the same as the square root of x?

Yes, for $x \ge 0$ . $x^{1/2} = \sqrt{x}$ because squaring it gives $x^1 = x$ .