Question

Find all complex numbers $z$ such that $z^4 = -4.$

Answer 1

Converting -4 to polar form gives $-4=4\exp(i\pi)$.

Then the 4th roots of -4 would be the numbers

$4^{1/4}\exp\left(i\dfrac{\pi+2k\pi}4\right)$

where k is taken from {0, 1, 2, 3}.

So we have

$z_1=4^{1/4}\exp\left(\dfrac{i\pi}4\right)=\sqrt2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)=1+i$

$z_2=\sqrt2\left(\cos\dfrac{3\pi}4+i\sin\dfrac{3\pi}4\right)=-1+i$

$z_3=\sqrt2\left(\cos\dfrac{5\pi}4+i\sin\dfrac{5\pi}4\right)=-1-i$

$z_4=\sqrt2\left(\cos\dfrac{7\pi}4+i\sin\dfrac{7\pi}4\right)=1-i$