Why Exponents Multiply

The exponents multiply because the product rule is applied to identical copies. $\left(x^{2}\right)^{4}$ is four copies of $x^{2}$ multiplied together: $x^{2}x^{2}x^{2}x^{2}$. All four copies have the same base, so the product rule adds the exponents: $2+2+2+2 = 8$. Adding the same number four times is the same as multiplying it by 4, which is $2\cdot 4$.

So the power of a power rule comes from the product rule. You apply the product rule to identical copies of a power, and the repeated addition becomes a multiplication.

The result is $\left(x^{2}\right)^{4} = x^{8}$. The same reasoning works for any exponents. If each copy holds $m$ factors and there are $n$ copies, there are $mn$ factors in all.