# Rational Exponents

How to solve rational exponents problems: formula, examples, and their solutions.

## Formula

The denominator *m* of a power means an *m*th root.

So *a*^{n/m} = ^{m}√*a*^{n}

## Example 1

4^{5/2} = 2^{2⋅(5/2)} = 2^{5}

The denominator 2 (blue) means a square root.

## Example 2

8^{-7/3}⋅8^{5/3} = 8^{-7/3 + 5/3}

Product of powers

The denominator 3 (blue) means a cube root.

8^{-2/3} = 2^{3⋅(-2/3)} = 2^{-2}

## Example 3

*x*^{-5/6} means 1/*x*^{5/6} = 1/(^{6}√*x*^{5}).

Negative exponent

To rationalize *x*^{5/6},

multiply *x*^{1/6}

to both of the numerator and the denominator.

Rationalizing a denominator

## Example 4

*x*^{2/3}/(*x*^{1/2}⋅*x*^{-1/6}) = *x*^{2/3}⋅*x*^{-1/2}⋅*x*^{1/6}

Negative exponent*x*^{2/3}⋅*x*^{-1/2}⋅*x*^{1/6} = *x*^{2/3 - 1/2 + 1/6}

Product of powers

## Example 5

*x*^{1/2} + 1 means √*x* + 1.

So rationalize *x*^{1/2} + 1 in the denominator.

Rationalizing a denominator