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Simplify. Assume all variables are positive. $\newline$ $(2r^{\frac{1}{3}})^8$ $\newline$ $\newline$ Write your answer in the form $A$ or $\frac{A}{B}$ , where $A$ and $B$ are constants or variable expressions that have no variables in common. All exponents in your answer should be positive. $\newline$ ______

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Power rule with rational exponents

Full solution

Q. Simplify. Assume all variables are positive.

\newline

(2r^{\frac{1}{3}})^8

\newline

\newline

Write your answer in the form

A

or

\frac{A}{B}

, where

A

and

B

are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.

\newline

______

Apply power rule: Use power of a product rule: $(a \cdot b)^n = a^n \cdot b^n$ . $(2r^{\frac{1}{3}})^8 = 2^8 \cdot (r^{\frac{1}{3}})^8$
Calculate $2^8$ : Calculate $2^8$ . $2^8 = 256$
Apply power of a power rule: Apply power of a power rule: $(a^n)^m = a^{(n \cdot m)}$ . $\newline$ $(r^{\frac{1}{3}})^8 = r^{(\frac{1}{3} \cdot 8)}$
Simplify $r^{\frac{8}{3}}$ : Simplify $r^{\left(\frac{1}{3}\right) \cdot 8}$ . $\newline$ $r^{\left(\frac{1}{3}\right) \cdot 8} = r^{\frac{8}{3}}$
Combine results: Combine the results from previous steps. $\newline$ $256 \times r^{\frac{8}{3}} = 256r^{\frac{8}{3}}$

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