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.
When you see a radical expression, 

The denominator of a fractional exponent indicates the root.
Specifically, = a ^{1/3} or in general: = 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.
0 out of 0 correct.