Definition of a Rational Exponent.

a ^m/n = ( ⁿ √ a ) ^m

In general, if _{$a^{1/n} = \sqrt[n]{a}$} is a real number, then

a ^{1/ n} = _{$a^{1/n} = \sqrt[n]{a}$}

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 _{$\sqrt[3]{8}$} 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 = ( a ² ) ^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 = 2 ² = 4

or

8 ^2/3 = (8 ² ) ^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, $\sqrt[3]{a} = a^{1/3}$ = a ^1/3 or in general: $a^{m/n} = (\sqrt[n]{a})^m$ = a ^{m / n}

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.