How to prove property of complex powers

Question

I have the following problem:

If $b$ is real, prove that $|a^b|=|a|^b$. In this case, $a$ is complex number.

I know the definition of a complex power, $a^b=e^{b\log(a)} $, but I´m not sure how to use it.

Answer 1

Let $a=|a|e^{i\theta}$. Then $$|a^b|=|e^{b \log a}|=|e^{b \log{|a|}+i\theta}|=|e^{b \log{|a|}}e^{i\theta}|=||a|^b|=|a|^b.$$

Answer 2

We can express $a$ as $r(\cos \theta + i\sin \theta)$. By De Moivre's Theorem, we have $$|a^b| = r^b|\cos b\theta + i\sin b\theta| = |a|^b.$$ The important part was that the absolute value of a complex number is multiplicative.