Exponent multiplication error

Question

$\left( x^{2/3}\right) ^{3/2}=x^{\dfrac {2}{3}\times \dfrac {3}{2}}=x^{1}$

Given this why is it that if I substitute $x=-1$ I get 1?:

$\left( \left( \sqrt [3] {-1}\right) ^{2}\right) ^{3/2}=\left( 1\right) ^{3/2}=1$

I feel I may have gone wrong somewhere but I can't see where..

Answer 1

Ina simplified version you are saying $$ ((-1)^2)^{1/2} = 1, $$ even though one expects $(x^2)^{1/2} = x$. Maybe you can figure it out now?

Answer 2

Your example shows that $a^t$ may be only properly defined for arbitrarely $t$ if $a\geq0$.