The Zero Exponent

The quotient rule leads to a new kind of exponent. $\frac{x^{7}}{x^{7}}$ is a nonzero number divided by itself, so its value is 1. The quotient rule says to subtract the exponents: $7-7 = 0$, which gives $x^{0}$.

Both answers describe the same expression, so they must be equal. For nonzero $x$, $x^{0} = 1$. The example shows that the zero exponent follows from the quotient rule.

You can also think about the factors. $x^{3}$ means three factors of $x$, and $x^{0}$ means no factors of $x$ at all. A product with no factors is 1, just as a sum with no terms is 0.