Q:

Find the value of z, $z^4 = -2$

I need the find the value of $z$ given the equation $$z^4 = -2$$
First I found that $$Arg(-2) = \pi +2\pi n$$
Since $$z = re^{iФ}$$ and $$z^4 = r^4e^{i4Ф}$$
$$r^4=2,  r = \sqrt[4]{2}$$
$$e^{i\pi} = e^{i4Ф}$$
which simplifies to $$\pi = 4Ф$$
We know that $$Ф = \frac{\pi}4+\frac{2\pi k}2$$
$$z = \sqrt[4]{2}(e^{i\frac{\pi}4+\frac{2\pi k}2})$$
$$z1 = \sqrt[4]{2}(cos(\frac{\pi}4)+isin(\frac{\pi}4))$$
$$z1 = \frac{\sqrt[4]{2^3}}2 + i\frac{\sqrt[4]{2^3}}2$$
And the values of a and b keep changing from positive to negative, you get a combination of 4 answers for z. Is this correct?  I asked this question in an equation solver and it gave me different values.  

A:

Your result seems correct.  You can test it writing the equation in the form:
$$
z^4+2=(z^2+i\sqrt{2})(z^2-i\sqrt{2})=0
$$
and solving:
$$
z^2+i\sqrt{2}=0 \qquad z^2-i\sqrt{2}=0
$$
the solutions are:
$$z=\pm\frac{\sqrt[4]{2^3}}{2}\pm i \frac{\sqrt[4]{2^3}}{2} $$

