What's $(-1)^{2/3}\; $? I know that 
$\left ( -1 \right )^{2/3}=\left ( \left ( -1 \right )^{2} \right )^{1/3}=1$
But Matlab computes this as $- 0.5 + 0.8660254038i$ a complex number.Why?

By convention, $-\frac12 + \frac{\sqrt3}2i \approx -0.5+0.866i$  is the principal value of $(-1)^\frac23$.
In particular, Matlab is presumably evaluating $(-1)^\frac23$ as $\exp(\frac23\log(-1))$ and choosing the principal branch of the logarithm, where $\log(-1) = \pi i$, and thus ending up with
$$\begin{aligned}
(-1)^\frac23
&= \exp\left(\tfrac23\log(-1)\right) \\
&= \exp\left(\tfrac23\pi i\right) \\
&= -\frac12+\frac{\sqrt3}2 i \\
&\approx -0.5+0.866i.
\end{aligned}$$
