1. The problem is to verify the equality $\frac{1}{x} = x^{-1}$. 2. Recall the rule of exponents: for any nonzero number $x$ and integer $n$, $x^{-n} = \frac{1}{x^n}$. 3. Applying this rule for $n=1$, we get $x^{-1} = \frac{1}{x}$. 4. Therefore, $\frac{1}{x} = x^{-1}$ is true by definition of negative exponents. 5. This means the expression on the left side and the right side represent the same value for any $x \neq 0$.