Find all complex number solutions for the equation $z^4=16$.

Step 1 ：First, we can write the equation in the polar form. In polar form, a complex number $z$ can be represented as $z=r(cos(\theta )+isin(\theta ))$, where $r$ is the modulus of $z$ and $\theta$ is the argument of $z$. In this case, $r=\sqrt{16}=4$ and $\theta =0$, so $z=4(cos(0)+isin(0))$.

Step 2 ：Next, we need to solve the equation $z^4=16$. Using De Moivre's theorem, we know that if $z=r(cos(\theta )+isin(\theta ))$, then $z^n=r^n(cos(n\theta )+isin(n\theta ))$. Therefore, we get $z^4=4^4(cos(4\times 0)+isin(4\times 0))=256$ which is not equal to 16. So, we need to find a new angle $\theta$ such that $z^4=16$. This gives us $4^4(cos(4\theta )+isin(4\theta ))=16$.

Step 3 ：Solving for $\theta$ gives us $\theta =\frac{k\pi }{2}$ for $k=0,1,2,3$. Substituting these values back into the polar representation of $z$ gives us the four complex roots of the equation $z^4=16$.