Question

Why is (`root(3)(-216))^5` equal to +7776 instead of negative?

(`root(3)(-216))^5`

Answer 1

`(root(3)(-216))^5 = -7776`

Explanation:

It is negative, not positive as your source seems to suggest.

The expression `root(3)(-216)` has two possible interpretations:

Real interpretation

As a real valued function of real numbers `f(x) = x^3` is one to one from `RR` onto `RR`. So the real cube root is also one to one from the whole of `RR` onto `RR`.

The real cube root of `-216`, which is `-6`, since `(-6)^3 = -216`.

Complex interpretation

As a complex valued function of complex numbers, `f(x) = x^3` is many to one, so we have to make a choice when we define what we mean by the principal cube root.

The principal complex cube root of `-216` is:

`6(cos (pi/3) + i sin (pi/3)) = 3+3sqrt(3)i`

Current example

Since this question is posted under Algebra, I guess I'm fairly safe in assuming that you intended the real cube root, in which case we find:

`(root(3)(-216))^5 = (-6)^5 = -7776`