Section 4: Rational Exponents

Definition of a Rational Exponent.

a^m/n  = (ⁿ√a)^m 

In general, if  is a real number, then

a^1/n =
 

The denominator of a fractional exponent is equal to the index of the radical. 

So 8^1/3  is the exponential form of the cube root of 8, and   is its radical form.

Next, we ask: what sense can we make of a symbol like a^2/3? According to the rules of exponents:

a^2/3 = (a^1/3)2  or a^2/3 = (a²)^1/3

 Example:

Use can use either of these rules to simplify the expression 8^2/3 as shown below.

Solution:

8^2/3 = (8^1/3)² = 2² = 4

or

8^2/3 = (8²)^1/3 = 64^1/3 = 4

Notice that we get the same answer either way. However, to evaluate a fractional power, it is more efficient to take the root first.

The denominator of a fractional exponent indicates the root.

Specifically,      or in general: 

Notice:   The index of the radical becomes the denominator of the rational power, and the exponent of the radicand (expression inside the radical) becomes the numerator. 

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