I need to calculate the following product:
$x^{-1} \cdot \sqrt[3]{x} = ?$
First, I apply the rule
$\sqrt[n]{x^m} = x^{\frac{m}{n}}$
to convert $\sqrt[3]{x}$ to $x^{\frac{1}{3}}$:
$x^{-1} \cdot \sqrt[3]{x} = x^{-1} \cdot x^{\frac{1}{3}}$
Then I add the exponents:
$x^{-1} \cdot \sqrt[3]{x} = x^{(-\frac{3}{3} + \frac{1}{3})}$
$x^{-1} \cdot \sqrt[3]{x} = x^{-\frac{2}{3}}$
I use the rule
$a^{-\frac{m}{n}} = \frac{1}{\sqrt[n]{a^m}}$
to transform $x^{(-\frac{2}{3})}$ to $\frac{1}{\sqrt[3]{x^2}}$.
However, if I check the solution by setting $x=1$ and calculating both sides, it turns out that it wrong.
Where exactly (in which of the above steps) did I make the error?