Absolute Value of Power

Theorem
Let $x$, $y$ be real numbers.

Let $x^y$, $x$ to the power of $y$, be real.

Then:


 * $\left \vert {x^y} \right \vert = \left \vert {x} \right \vert ^y$

Proof
If $x = 0$, the theorem clearly holds, by the definition of powers of zero.

Suppose $x \ne 0$.

We use the interpretation of real numbers as wholly real complex numbers.

Likewise we interpret the absolute value of $x$ as the modulus of $x$.

Then $x$ can be expressed in polar form:


 * $x = r e^{i\theta}$

...where $r = \left \vert {x}\right \vert$ and $\theta$ is an argument of $x$.

Then: