Find all fourth roots of -4 , and identify the one with…

Find all fourth roots of $-4$, and identify the one with smallest positive argument.

Step 1 · Polar form of $-4$

$$ -4=4\left(\cos\pi+i\sin\pi\right)=4e^{i\pi} $$

Step 2 · Apply the $n$-th root formula

The fourth roots of $4e^{i\pi}$ are:

$$ z_k=4^{1/4}\exp\left(i\,\frac{\pi+2k\pi}4\right)=\sqrt2\,\exp\left(i\,\frac{(2k+1)\pi}4\right), \qquad k=0,1,2,3 $$

giving arguments $\dfrac\pi4,\ \dfrac{3\pi}4,\ \dfrac{5\pi}4,\ \dfrac{7\pi}4$, all with modulus $\sqrt2$.

Step 3 · The root with smallest positive argument

The smallest positive argument is $\dfrac\pi4$ ($k=0$):

$$ z_0=\sqrt2\left(\cos\frac\pi4+i\sin\frac\pi4\right)=\sqrt2\left(\frac{\sqrt2}2+i\frac{\sqrt2}2\right)=1+i $$

Step 4 · Verify by direct computation

$$ (1+i)^2=1+2i-1=2i, \qquad (1+i)^4=(2i)^2=-4 \quad\checkmark $$

Step 5 · The full set of roots

$$ \boxed{z_0=1+i,\quad z_1=-1+i,\quad z_2=-1-i,\quad z_3=1-i} $$

(the four vertices of a square of side $2$ centered at the origin) — and the root with smallest positive argument is $z_0=1+i$.