A rational exponent is an exponent written as a fraction.

Example:

x^{1/2}, 8^{2/3}, y^{3/4}​

“Rational” just means the exponent is a ratio of two integers (a fraction).

The BIG idea (most important rule)

a^{m/n} = sqrt(^[n] {a^m})​

This rule says:

• the denominator = the type of root

• the numerator = the power

Breaking it down (slow and clear)

1. The denominator tells you the root

a^{1/2} = sqrt{a}

a^{1/3} = sqrt(^[3]{a})

a^{1/4} = sqrt(^[4] {a})​

2. The numerator tells you the power

a^{2/3} = sqrt(^[3] {a^2})​

Examples (numbers first)

Example 1:

9^{1/2} = sqrt{9} = 3

Example 2:

8^{2/3}​

Step 1: Cube root (denominator = 3)

sqrt(^[3] {8^2})​

Step 2: Power first or root first (either works):

sqrt(^[3] {64}) = 4

Example 3:

16^(3/4)

sqrt(^[4] {16^3}) = sqrt(^[4] {4096}) = 8

Rational exponents with variables

Example:

x^(3/2)

sqrt{x^3} = sqrt{x^2 * x} = x*sqrt{x}​

Another:

y^{1/3} = sqrt{^[3] {y}}​

Why use rational exponents instead of radicals?

Because:

• exponent rules still work

• expressions are easier to manipulate

• advanced algebra expects this form

Example:

x^{1/2} * x^{3/2} = x^2

(Just add exponents!)

Important rules still apply

All exponent laws work with rational exponents:

• a^m a^n = a^{m+n}

• (a^m)^n = a^{mn}

• a^(−m) = 1/a^(m)​