Finding the value of $(-1+i)^{10}$ by using the trigonometric form's expression I am trying to find out what is the value of $(-1+i)^{10}$ by using the trigonometric expression, which should simplify it by avoiding to multiply in binomial form. So far I got this (sin==sen, and excuse my shoddy writing):



But every other calculator I used to test the result says that it's -$32i$, and not $32i$, which is driving me mad! I can only guess that I messed u on the argument($\alpha$)

Any help is welcome!

Edit: Solved, I hadn't added the fractions properly, see comment chain below post
$$-1+i=\sqrt2\exp(3\pi i/4)$$
so
$$(-1+i)^{10}=32\exp(15\pi i/2)=-32i.$$
Alas,
$$\sqrt2\exp(5\pi i/4)=-1-i.$$
Hint:

Use https://en.m.wikipedia.org/wiki/Atan2#Definition_and_computation

arg$(-1+i)=\arctan\dfrac1{-1}+\pi=\dfrac{3\pi}4$

arg$(-1+i)^{10}=10\cdot\dfrac{3\pi}4=\equiv-\dfrac\pi2\pmod{2\pi}$