Why can't a negative number be raised to a power?

Question

I was messing around with graphs on desmos, and I found when I put in the function (n)^x, when n was negative, there would be no graph. Is there a reason for this? https://www.desmos.com/calculator/19j6f3ss8s

Answer 1

You seem to be operating under a misconception: it is possible to raise a negative number to a nonzero power. For instance, $$(-2)^3=(-2)\times(-2)\times(-2)=-8.$$ It is true that some exponents present problems: since we can't take square roots of negative numbers, any exponent with a "$2$" in the denominator, like $$(-2)^{3\over 4},$$ causes problems. For such cases (and in general, to make sense of negative numbers raised to irrational powers, like $(-2)^\pi$) we turn to complex numbers.

But that's another story, and the main point for now is that your assumption is wrong: we can indeed raise negative numbers to lots of powers.

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Your edit makes things clearer. The problem with graphing, say, $y=(-2)^x$ is that it's undefined a lot of the time - so frequently so, that it doesn't even make sense to graph it!

First, let's think about rational values of $x$. If $x={p\over q}$ (in lowest terms) with $q$ even, then computing $(-2)^{x}$ requires us to take the square root of a negative number, which we can't do in the real numbers. The problem is, such "bad" fractions occur densely in the real numbers: if $a<b$, then there is some such $x$ in between $a$ and $b$. So any graph will have lots of "holes".

It gets worse: what about irrational values? We define, say, $3^\pi$ as the limit of $3^q$ for rational $q$, as $q\rightarrow \pi$ (this takes some work to make sense of). But since there are lots of "bad" values near $\pi$, this approach is problematic for computing $(-2)^\pi$.

We can make things much better by working with complex numbers, instead. But, inside the context of the real numbers, $(-2)^x$ fails to be defined for so many values of $x$ that it's better not to graph it at all.