Question

How do you find all the real cube roots of 8/125?

Answer 1

`8/125` has one Real cube root `2/5`

Explanation:

`8 = 2^3` and `125 = 5^3`

So

`root(3)(8/125) = root(3)(2^3/125^3) = root(3)(2^3)/root(3)(5^3) = 2/5`

The other two roots are both Complex:

`2/5 omega = 2/5((-1+i sqrt(3))/2) = (-1+i sqrt(3))/5`

`2/5 omega^2 = 2/5((-1+i sqrt(3))/2) = (-1-i sqrt(3))/5`

where `omega = (-1+i sqrt(3))/2` is the primitive Complex cube root of unity.