$(-1)^3$ has different results when evaluated as $(-1)\times(-1)\times(-1) = -1$ vs $((-1)^2)^{3/2} = 1$. Which is correct? I know that 
$$(-1)^3=(-1)\times(-1)\times(-1)=-1 \tag{1}$$ 
but also 
$$(-1)^3=((-1)^2)^{3/2}=1^{3/2}=1 \tag{2}$$
So which gives the correct value of $(-1)^3$?

The first equality is correct. The second one isn't: it assumes that $a^{bc}=(a^b)^c$. This is true indeed if $a>0$ (and $b,c\in\mathbb R$), but $-1<0$.
