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:

  1. 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.
  2. 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.

rationalpower3green lineFor 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!


The explanation for this surprising result is not brief, but I discussed it in a previous series of posts:

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