What is the value of $z$? $ z^4 = -1$

Question

how to calculate the value of Z ?

$$z^4 = -1$$

Can anyone help me with this problem here.

Answer 1

$$z^4=e^{i\pi}=e^{(2n+1)i\pi}$$ where $n$ is any integer

$$\implies z=e^{(2n+1)i\pi/4}$$ where $n\equiv0,1,2,3\pmod4$

We can optionally use How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?

Answer 2

The equation $z^4=-1$ can be rewritten as $z^4+1=0$ and this in turn can be written as $$z^4+2z^2+1-2z^2=(z^2+1)^2-(\sqrt 2 z)^2=(z^2+\sqrt 2 z+1)(z^2-\sqrt 2 z+1)=0$$

And the solutions you want are solutions of the two quadratics $$z^2+\sqrt 2 z+1=0$$and$$z^2-\sqrt 2 z+1=0$$

Answer 3

The hint: $$Z=\sqrt[4]{-1}=\sqrt[4]{1\operatorname{cis}\pi}=\operatorname{cic}\frac{\pi+2\pi k}{4},$$ where $k\in\{0,1,2,3\}$.

Also, you can use the following way.

Since $$Z^4+1=Z^4+2Z^2+1-2Z^2=(Z^2+\sqrt2Z+1)(Z^2-\sqrt2Z+1),$$ we obtain: $$\left\{\pm\frac{1}{\sqrt2}\pm\frac{1}{\sqrt2}i\right\}.$$