Question

Find all the
cube roots of 8​

Answer 1

Explanation:

Cuberoot of 8

2*2*2=2

Answer 2

Since you're mentioning "all" the cube roots, I'm assuming you're asking about complex roots as well.

In that case, it is more convenient to use De Moivre's form:

`z=8=8(1+0i)=8[\cos(2\pi)+i\sin(2\pi)]`

Which implies

`\sqrt[3]{z}=2\left[\cos\left((2n\pi)/(3)\right)+i\sin\left((2n\pi)/(3)\right)\right],\quad n=0, 1, 2`

Which yields the roots

`z_0 = 2\left[\cos\left(0\right)+i\sin\left(0\right)\right] = 2`

`z_1=2\left[\cos\left((2\pi)/(3)\right)+i\sin\left((2\pi)/(3)\right)\right]=-1+i√(3)`

`z_2=2\left[\cos\left((4\pi)/(3)\right)+i\sin\left((4\pi)/(3)\right)\right]=-1-i√(3)`