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Simplify. Express your answer using a single exponent. $\newline$ $(3w^3)^4$

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Powers of a power: variable bases

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Q. Simplify. Express your answer using a single exponent.

\newline

(3w^3)^4

Apply Power Rule: Apply the power of a power rule to the expression $(3w^3)^4$ . The power of a power rule states that when raising a power to another power, you multiply the exponents. In this case, we have an exponent of $4$ being applied to both $3$ and $w^3$ . $(3w^3)^4 = 3^4 \times (w^3)^4$
Calculate $3^4$ : Calculate $3^4$ . $\newline$ $3^4$ means $3$ multiplied by itself $4$ times. $\newline$ $3^4 = 3 \times 3 \times 3 \times 3 = 81$
Calculate $(w^3)^4$ : Calculate $(w^3)^4$ . Using the power of a power rule, we multiply the exponents $3$ and $4$ . $(w^3)^4 = w^{(3*4)} = w^{12}$
Combine Results: Combine the results from Step $2$ and Step $3$ . $\newline$ We have found that $3^4 = 81$ and $(w^3)^4 = w^{12}$ . Now we combine these results to express the simplified form of the original expression. $\newline$ $(3w^3)^4 = 81 \times w^{12}$

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