Inverse Functions: Rational Exponents (Part 8)

In this series of posts, we have seen that the definition of $\sqrt[n]{x}$ and saw that the definition was a little different depending if $n$ is even or odd:

If $n$ is even, then $y = \sqrt[n]{x}$ means that $x = y^n$ and $y \ge 0$ . In particular, this is impossible (for real $y$ ) if $x < 0$ .
If $n$ is odd, then $y = \sqrt[n]{x}$ means that $x = y^n$ . There is no need to give a caveat on the possible values of $y$ .

Let’s now consider the definition of $x^{m/n}$ , where $m$ and $n$ are positive integers greater than 1. Ideally, we’d like to simply defined

$x^{m/n} = \left[ x^{1/n} \right]^m$

This definition reduces to previous work (like a good MIT freshman), using prior definition for raising to powers that are either integers or reciprocals of integers. Indeed, if $x \ge 0$ , there is absolutely no ambiguity about this definition.

Unfortunately, if $x < 0$ , then a little more care is required. There are four possible cases.

Case 1. $m$ and $n$ are odd. In this case, there is no ambiguity if $x < 0$ is negative. For example,

$(-32)^{3/5} = \left[ (-32)^{1/5} \right]^3 = [-2]^3 = -8$

Case 2: $m$ is even but $n$ is odd. Again, there is no ambiguity if $x< 0$ is negative. For example,

$(-32)^{4/5} = \left[ (-32)^{4/5} \right]^3 = [-2]^4 = 16$

Case 3: $m$ is odd but $n$ is even. In this case, $x^{m/n}$ is undefined if $x < 0$ . For example, we would like $(-16)^{3/2}$ to be equal to $\left[ (-16)^{1/2} \right]^3$ , but ${-16}^{1/2} = \sqrt{-16}$ is undefined (using real numbers).

Case 4. $m$ and $latex $n$ are both even. This is perhaps the most interesting case. For example, how should we evaluate $(-8)^{4/6}$ ?. There are two legitimate choices… which lead to different answers!

Option #1: If we just apply the proposed definition of $x^{m/n}$ , we find that

$(-8)^{2/6} = \left[ (-8)^2 \right]^{1/6} = [64]^{1/6} = 2$

Option #2: We could first reduce $2/6$ to lowest terms:

$(-8)^{2/6} = (-8)^{1/3} = -2$

So… which is it?!?!?!?! The rule that mathematicians have chosen is that simplifying the exponent takes precedence over the above definition. In other words, the definition $x^{m/n} = \left[ x^{1/n} \right]^m$ should only be applied in $m/n$ has been reduced to lowest terms in order to remove the above ambiguity.

For the sake of completeness, I note that the above discussion restricts our attention to real numbers. If complex numbers are permitted, then things become a lot more interesting. If we repeat a few of the above calculations using complex numbers, we get answers that are different!