All Three Rules Together

Many expressions use more than one exponent rule. $\left(x^{2}\right)^{3}x^{4}$ has a nested power next to a plain power. Work from the inside out. First simplify the nested power: $\left(x^{2}\right)^{3} = x^{6}$. Then use the product rule on the two powers side by side: $x^{6}x^{4} = x^{10}$.

Inside out is a reliable order. Simplify each set of parentheses to a single power, then combine what is left with the product and quotient rules. Name the rule before you use it, one rule per step.

Different orders of the rules can lead to the same answer. For example, $x^{6}x^{4}$ and $\left(x^{2}\right)^{5}$ both simplify to $x^{10}$. Use each rule only where its conditions hold. The product rule needs the same base, and the power of a power rule needs a power inside parentheses.