Rational Exponents

Rational Exponents

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

Formula

The denominator m of a power means the mth root.

The denominator m of a power means an mth root.

So an/m = man

Example 1

Simplify the given expression. 4^(5/2).

45/2 = 22⋅(5/2) = 25

The denominator 2 (blue) means a square root.

Example 2

Simplify the given expression. (8^(-7/3))(8^(5/3)).

8-7/3⋅85/3 = 8-7/3 + 5/3

Product of powers

The denominator 3 (blue) means a cube root.

8-2/3 = 23⋅(-2/3) = 2-2

Negative exponent

Example 3

Simplify the given expression. x^(-5/6).

x-5/6 means 1/x5/6 = 1/(6x5).

Negative exponent

To rationalize x5/6,
multiply x1/6
to both of the numerator and the denominator.

Rationalizing a denominator

Example 4

Simplify the given expression. (x^(2/3)) / ((x^(1/2))(x^(-1/6))).

x2/3/(x1/2x-1/6) = x2/3x-1/2x1/6

Negative exponent

x2/3x-1/2x1/6 = x2/3 - 1/2 + 1/6

Product of powers

Example 5

Simplify the given expression. (x^(1/2) - 3) / (x^(1/2) + 1).

x1/2 + 1 means √x + 1.

So rationalize x1/2 + 1 in the denominator.

Rationalizing a denominator

FOIL method