What is wrong in my calculation ($x^{-1} cdot sqrt[3]{x} = ?$)?

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?