Complex Roots with improper fraction I'm having trouble with the following:
$(-16i)^{5/4}$
My calculations for the Principal root is:
$32(\cos (3\pi/2) * 5/4) + i \sin (3\pi/2)* 5/4))$
$=32(Cis (15\pi/8))$
This answer does not agree with the online calculators.  It gives a positive real value and the online calculators show a principal angle in Quadrant 3.
Confused on what happened here.

First write $-16i$ in general polar form: $16e^{3\pi i/2+2\pi ik}$. Then $$(16e^{3\pi i/2+2\pi ik})^{5/4}=32e^{15\pi i/8+5\pi ik/2}=32e^{15\pi i/8+20\pi ik/8}.$$
By playing with values of $k$, we obtain the separate roots $32e^{\frac{3+4k}{8} \pi i}$.
Now, the discrepancy here is which of these is actually the principal root. I believe this is a matter of definitions. According to Wikipedia, the principal root is the one with the smallest positive argument. By that definition the answer has argument $3\pi/8$. However, I believe that Wolfram (and your calculator) has calculated the principal root by beginning with the smallest absolute value argument, which gives $16e^{-\pi i/2}$, and raising to the $5/4$, giving $32e^{-5\pi i /8}=32e^{11\pi i/8}$, which I believe is the answer you are looking for.
