# Rational Exponents

Definition â€” Rational Exponent With Numerator m

If a is a real number, m is a nonnegative integer, n is a positive integer, and is in lowest terms, then

If n is even, then am/n is a real number only when a 0.

Examples:

Note:

When m = 1, we have

That is

Example 1

Rewrite using a radical and evaluate: 323/5

Solution

We will use three different methods to solve this problem.

 Method 1 Use 323/5 Here, a = 32, m = 3, and n = 5. Find the prime factorization of 32. 32 = 2 Â· 2 Â· 2 Â· 2 Â· 2 = 25. Simplify the radical. Evaluate 23. = (2)3= 8 Method 2 Use Comparing am/n to 323/5 we see that a = 32, m = 3, and n = 5. Use your calculator to find 323. Use your calculator to find = 8 Method 3 Use the Power of a Power Property of Exponents, (am)n = amn. 323/5 Find the prime factorization of 32. 32 = 2 Â· 2 Â· 2 Â· 2 Â· 2 = 25. = (25)3/5 Use the Power of a Power Property of Exponents. Multiply the exponents and write the result in lowest terms. Evaluate 23. = 23 = 8

In summary, all three methods lead to the same result: 323/5 = 8

Note:

Method 2 typically means dealing with larger numbers since we first evaluate the power.