Rational Exponents

We can also use exponential notation to represent a radical. To do so, we use a rational exponent.

Recall that a rational number is a number that may be written as the ratio of two integers.

So, a rational exponent may be a fraction, such as , or an integer such as .

Definition — Rational Exponent With Numerator 1

If a is a real number, then

Here, n is a positive integer.

If n is odd, then a^1/n is a real number.

If n is even, then a^1/n is a real number only when a ≥ 0.

Using this definition, we write:

Note:

The way we interpret 2n depends upon the value of n:

• n is a whole number

• n is zero 2⁰ = 1

• n is a negative integer

• n is a fraction

Example

Rewrite using a radical and evaluate: 125^1/3

Solution

Write the radical symbol.
The index is 3, the denominator of the exponent
The radicand is 125, the base of the exponential expression.

Find the prime factorization of 125.

125 = 5 · 5 · 5 = 5³.

Since 5³ = 125, the cube root of 125 is 5.

So,