Question

Express −1+i in polar form. Hence show that (−1+i)16 is real and that 1/(−1+i)6 is purely imaginary, giving the value for each.

Answer 1

$-1+i=\sqrt{2}(-\frac{\sqrt{2}}2+i\frac{\sqrt{2}}2)=\sqrt{2}(cos\frac{3\pi}4+i \,sin\frac{3\pi}4)$ .

Applying de Moivre's formula we get:

$(-1+i)^{16}= 2^8(cos(12\pi)+i \,sin(12\pi))=256$ ,

$(-1+i)^{6}= 2^3(cos\frac{9\pi}2+i \,sin\frac{9\pi}2)=8i$ ,

$\frac{1}{(-1+i)^{6}}=-\frac{1}8i$ .