Evaluating fractional exponents

An SAT Math micro-topic under Radicals and rational exponents (Advanced Math). Free to read — no account needed.

A fractional exponent combines a power and a root: in

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

, the denominator

n

is the root and the numerator

m

is the power. So

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

is the

n

th root of

a

, raised to the

m

th power.

To find

27^{\frac{2}{3}}

, take the cube root first,

\sqrt[3]{27} = 3

, then raise to the power

2

3^2 = 9

. Doing the root before the power keeps the numbers small.

A negative sign in the exponent means take the reciprocal.

16^{-\frac{1}{2}} = \frac{1}{16^{\frac{1}{2}}} = \frac{1}{\sqrt{16}} = \frac{1}{4}

When the variable is the base, raise both sides to the reciprocal of the exponent. From

m^{\frac{2}{3}} = 9

, the reciprocal of

\frac{2}{3}

\frac{3}{2}

, so

m = 9^{\frac{3}{2}} = (\sqrt{9})^3 = 27

Worked examples

What is

16^{\frac{3}{4}}

?
Take the fourth root:

\sqrt[4]{16} = 2

.
Raise to the power

3

2^3 = 8

What is

8^{-\frac{1}{3}}

?
The negative sign means reciprocal:

8^{-\frac{1}{3}} = \frac{1}{8^{\frac{1}{3}}}

.
Since

\sqrt[3]{8} = 2

, this equals

\frac{1}{2}

Solve

m^{\frac{2}{3}} = 4

for

m

.
Raise both sides to the reciprocal

\frac{3}{2}

m = 4^{\frac{3}{2}}

.
So

m = (\sqrt{4})^3 = 2^3 = 8

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