When and why is $x^x$ undefined?

Indeed, as you mentioned, something like $(-10)^{-10}$ is just $1/(-10)^{10}$.

The problem is that anywhere that we have rationals we will be very close to a value of $x$ which is of the form $-\frac{1}{2}p$.

In other words, note that $(-1/2)^{-1/2}$ is equal to $\frac{1}{(-1/2)^{1/2}}$ and we cannot take the square root of a negative number.

Likewise $$(-1/4)^{-1/4}=\frac{1}{(-1/4)^{1/4}}=\frac{1}{((-1/4)^{1/2})^{1/2}}$$ and again we have the square root of a negative number.

It turns out we have infinitely many of these cases along the negative real number line. Even though we technically can let the domain have some points, such as $-10$ or $-1$, it is easier if we just say the domain is $(0,\infty)$.

Tags:

Analysis

Calculus

Algebra Precalculus

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