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Simplify. Express your answer using a single exponent. $\newline$ $(2z^3)^3$

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Powers of a power: variable bases

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

\newline

(2z^3)^3

Apply Power Rule: Apply the power of a power rule to $(2z^3)^3$ . The power of a power rule states that when raising a power to another power, you multiply the exponents. For a term $(ab)^n$ , where $a$ is a coefficient and $b$ is a variable with an exponent, the rule is applied to both the coefficient and the variable separately. $(2z^3)^3 = 2^3 \times (z^3)^3$
Calculate $2^3$ : Calculate $2^3$ . $\newline$ $2^3 = 2 \times 2 \times 2 = 8$
Calculate $(z^3)^3$ : Calculate $(z^3)^3$ . $(z^3)^3 = z^{(3*3)} = z^9$
Combine Results: Combine the results from Step $2$ and Step $3$ . $\newline$ From Step $2$ , we have $2^3 = 8$ , and from Step $3$ , we have $(z^3)^3 = z^9$ . Combining these gives us the final simplified expression: $\newline$ $(2z^3)^3 = 8z^9$

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