Why is this not the correct way to treat this exponent? [closed]
If $x$ is a positive real number, then \begin{align*} x^{a + b} & = x^ax^b\\ (x^a)^b & = x^{ab} \end{align*} Therefore, the step $$x^{-1 \cdot \frac{2}{3}} = x^{-1}x^{2/3}$$ is false. $$x^{-2/3} = x^{-1 \cdot \frac{2}{3}} = \left(x^{2/3}\right)^{-1} = \frac{1}{x^{2/3}}$$ while $$x^{-1}x^{2/3} = x^{-1 + \frac{2}{3}} = x^{-1/3} = \left(x^{1/3}\right)^{-1} = \frac{1}{x^{1/3}}$$