Firstly I am not a student of mathematics just because right now I am doing a course related to complex variable I am having interest on learning things in a proper way .
Suppose, $Z=1+i$ and $C=1-i$
what is the value of $|Z^C|$ ?
Using principle value of $z^c$:=$exp(c logz)$
First i have found the value of logz for which i have used the formula $logz=lnr+i(argz)$.
After simplification ,
$exp(ln\sqrt2 +\frac{\pi}{4}+i\frac{\pi}{4}-iln\sqrt2)$
What should be the next step ? or is there any other way to solve the problem ?
Since $e^{x+iy}=e^xe^{iy}$ and we only care about the modulus, we ignore the imaginary parts of the exponent. The answer is $e^{\ln\sqrt2+\pi/4}=\sqrt2e^{\pi/4}$.