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Fractional exponents

A fraction in the exponent mixes a root and a power. The denominator says which root to take, and the numerator says which power. You may take the root first or the power first, and you choose the order that gives the neatest numbers. The power rules still hold: exponents are added when you multiply the same base, and subtracted when you divide.

Try it yourself

Exponential growth

a^{1/n} = \sqrt[n]{a}

a root as an exponent

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

root and power

\frac{a^m}{a^n} = a^{m-n}

division of powers

Symbols

$m$	the power
$n$	the root

Example

$8^{2/3} = 4$ , since $\sqrt[3]{8} = 2$ and $2^2 = 4$ . The exponent in $\frac{a^3\cdot a^{-1}}{a^{1/2}}$ is $3 - 1 - 0.5 = 1.5$ .

Take the root first when the base is a perfect cube or square.

Practise powers, roots and logarithms for free →

← Polynomial division · Simplifying square roots →

Part of Mathematics 1T: Powers, roots and logarithms.