Power rule for modulus and complex numbers

Question

If $z \in \mathbb{C}$ then is \begin{equation} |z|^{1/2} = |z^{1/2}| \end{equation} true? Note: $| z |$ denotes the modulus of $z$.

I know this is true if $z$ is real and I have tried some examples in the complex case and they seem to work but is there a proof for any $z$? Also, what about \begin{equation} |z|^n = |z^n|? \end{equation}

Is there only certain values of $n$ or $z$ that satisfy this?

Answer 1

By exponential form $z=\rho e^{i\theta}$ with $\rho\ge 0$ we have

• $|z|=\rho \implies |z|^\frac12 =\rho^\frac12$
• $z^\frac12 =\rho^\frac12e^{ik\frac \theta 2},\;k=0,1 \implies |z^\frac12|=\rho^\frac12|e^{ik\frac \theta 2}|=\rho^\frac12$

in a similar way we can prove the case $|z|^n=|z^n|$.