Positive Rational Exponents

When a rational number is used as an exponent, the denominator indicates “root” and the numerator indicates “power.” For example, 8^2/3 means (8^1/3)². We take the cube root of 8 to get 2, then square 2 to get 4. Thus 8^2/3 = 4.

In general, a^m/n is the mth power of the nth root of a. Rational Exponents If m and n are positive integers, then a^m/n = (a^1/n)^m, provided that a^1/n is defined.

Example 1

Evaluating expressions with positive rational exponents

Evaluate each expression.

a) 16^3/4

b) 27^4/3

c) (-8)^2/3

d) -4^3/2

Solution

a) To evaluate 16^3/4, take the fourth root of 16 to get 2 and then cube 2 to get 8. These steps are written as follows:

16^3/4 = (16^1/4)³ = 2³ = 8

b) To evaluate 27^4/3, take the cube root of 27 to get 3, and then raise 3 to the fourth power to get 81:

27^4/3 = (27^1/3)⁴ = 3⁴ = 81

c) (-8)^2/3 = [(-8)^1/3]² = (-2)² = 4

d) -4^3/2 = -(4^1/2)³ = -2³ = -8

Caution

We can evaluate -4^3/2 because the negative sign is saved until last. But (-4)^3/2 is not a real number because (-4)^1/2 is not a real number.

By the definition of rational exponents we find the root and then the power. For example,

8^2/3 = (8^1/3)² = 2² = 4.

However, we get the same result if we find the power and then the root:

8^2/3 = (8²)^1/3 = 64^1/3 = 4

In general, a^m/n is also the nth root of the mth power of a.

Evaluating a^m/n in Either Order

If m and n are positive integers, then a^m/n = (a^1/n)^m = (a^m)^1/n, provided that a^1/n is defined.

The fact that the power can be found before the root will be used later in this chapter. It is not useful in simply evaluating expressions because if the power is evaluated first, we might have to find the root of a very large number. For example, finding the power first in Example 1(b), we get

27^4/3 = (27⁴)^1/3 = 531,441^1/3

Calculator Close-up

It does not matter whether you use a fraction or a decimal as the exponent as long as the decimal is the exact value of the fraction. You will get the same result whether you find the root first or the power first.

Helpful hint

Note that in a^m/n we do not require m/n to be reduced. As long as the nth root of a is real, then the value of a^m/n is the same whether or not m/n is in its lowest terms.