Radicals and rational exponents

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Radicals and exponents are two ways of writing the same thing.
A root becomes a fractional exponent:

\sqrt{x} = x^{1/2}

\sqrt[3]{x} = x^{1/3}

, and in general

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

When the exponent is a fraction

\frac{m}{n}

, the bottom is the root and the top is the power:

x^{m/n} = \sqrt[n]{x^m}

.
You can also take the root first:

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

Switching to whichever form is easier often simplifies the work.
To evaluate

27^{2/3}

, take the cube root first:

\sqrt[3]{27} = 3

, then square it to get

9

A negative rational exponent means take the reciprocal.
For example,

x^{-1/2} = \frac{1}{x^{1/2}} = \frac{1}{\sqrt{x}}

Worked examples

Write

\sqrt[3]{x^2}

as a power of

x

.
The bottom of the exponent is the root (

3

) and the top is the power (

2

).
So

\sqrt[3]{x^2} = x^{2/3}

Evaluate

16^{3/4}

.
Take the fourth root first:

\sqrt[4]{16} = 2

, then raise it to the third power.
So

16^{3/4} = 2^3 = 8

Write

x^{5/2}

as a radical in simplest form.
The bottom

2

is a square root and the top

5

is the power:

x^{5/2} = \sqrt{x^5}

.
Since

x^5 = x^4 \times x

, this simplifies to

x^2\sqrt{x}

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