Product Rule or Power Rule?

The product rule and the power of a power rule are easy to mix up. The product $x^{4}x^{3}$ multiplies two different powers side by side, so the exponents add and the result is $x^{7}$. The power $\left(x^{4}\right)^{3}$ makes three copies of one power, so the exponents multiply and the result is $x^{12}$.

To choose the rule, look at how the expression is written. Two powers side by side with a multiplication sign between them means you add the exponents. One power inside parentheses with an exponent outside means you multiply the exponents. Decide which form you have before you touch the exponents.

If you are not sure, write out a small case. $x^{2}x^{2}$ has four factors of $x$, and $\left(x^{2}\right)^{2}$ also has four. The two rules give the same answer here only because $2+2$ and $2\cdot 2$ are both 4.