Question

What are all the cube roots of $-64$?

Answer 1

Determine the cube roots of $-64$

Find the required roots:

$$\begin{gathered} z^3=-64 \\ \Rightarrow z^3+64=0 \\ \Rightarrow z^3+4^3=0 \\ \Rightarrow \left(z+4\right)\left(z^2+4^2-4z\right)=0 \\ \Rightarrow \left(z+4\right)=0,\mathrm{or}\left(z^2+4^2-4z\right)=0 \\ \Rightarrow z=-4,\mathrm{or}\left(z^2-4z+16\right)=0 \end{gathered}$$

Solve$\left(z^2+4^2-4z\right)=0$:

$$\begin{gathered} \left(z^2-4z+16\right)=0 \\ \Rightarrow z=\frac{4\pm \sqrt{4^2-4\times 1\times 16}}{2\times 1} \\ \Rightarrow z=\frac{4\pm \sqrt{16-64}}{2} \\ \Rightarrow z=\frac{4\pm \sqrt{-48}}{2} \\ \Rightarrow z=\frac{4\pm 4\sqrt{3}i}{2} \\ \Rightarrow z=\frac{4+4\sqrt{3}i}{2},\frac{4-4\sqrt{3}i}{2} \\ \therefore z=2+2\sqrt{3}i,2-2\sqrt{3}i \end{gathered}$$

Hence, the cubic roots of $-64$ are $-4,2+2\sqrt{3}i,2-2\sqrt{3}i$.