Is that true $(z^{c_1})^{c_2} =z^{c_1c_2}$?

Question

Is that true $(z^{c_1})^{c_2} =z^{c_1c_2}$ for any complex number $c_1, c_2$ and $z \neq 0$? I have computed it by expressing $z, c_1, c_2$ in standard form and by the definition of power function and I think it is true, but I can not believe my complicated calculation.

Answer 1

It is a good thing you don’t believe it, because it is not true. Just consider $$-1=(-1)^1=(-1)^{2\cdot\frac12}\neq\big((-1)^2\big)^{1/2}=1^{1/2}=1.$$