What if we have a fractional exponent that is an improper fraction? For example, what if we have the problem 4^3/2? Keep in mind that 3/2 is an improper fraction whose mixed number counterpart is 1.5. When we turn this improper fraction into a mixed number, we make the fraction component of the mixed number, as well as the numerator of the improper fraction, our exponents. All this means, for this example, is that the number 4 is raised to the 1/2 power and the 3rd power, in any order. If we raise it to the 3rd power, we get 64. Then we can raise it to the 2nd power.
We can follow the rules we learned before about how the denominator of a fractional component constitutes the index of a radical expression. Therefore, we can place the 64 under a square root sign and place the denominator to the upper left of the square root sign as its index. The number will be 2. Since 2 is always implied in ordinary radical expressions with no index (hence, 'square' root), we don't need to actually explicitly put the 2 in the upper left-hand corner. We simply take for granted that it is the '2nd' or square root. This places us with a solution of 8, since 8 is the square root of 64. Keep in mind that we could have either found the 1/2 root or the 3rd root first and we would have gotten the same solution.