A problem in algebra: how does $-1=1$?

Question

I have algebra problem from a friend, that is 1=-1!!! because

$$-1=-1^{3}=-1^{^{\frac{6}{2}}}=\sqrt{(-1)^6}=\sqrt{1}=1$$

I can not see what is wrong with this? I will appreciate it any help.

Answer 1

The usual properties of roots/exponents apply when the basis is (real) positive. In this case, we can't have

$$\;-1^3=(-1)^3=(-1)^{6/2}\color{red}{\stackrel {!!}=}\left[(-1)^{1/2}\right]^6$$

as $\;(-1)^{1/2}=\sqrt{-1}\;$ cannot be done within the real numbers (and this is also another reason why the above mentioned properties don't apply to complex numbers...)