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Find the value of the following. a) $\sqrt[3]{375} \times \sqrt[3]{192}$

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Evaluate rational exponents

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Q. Find the value of the following. a)

\sqrt[3]{375} \times \sqrt[3]{192}

Identify numbers: Identify the numbers to find the cube roots for. Calculate the cube root of $375$ and the cube root of $192$ .
Simplify using prime factorization: Simplify the cube roots using prime factorization. $375 = 3 \times 5 \times 5 \times 5 = 3 \times 5^3$ , $192 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^6 \times 3$ .
Calculate cube roots: Calculate the cube roots. $\newline$ Cube root of $375$ = cube root of $(3 \times 5^3)$ = $5 \times$ cube root of $3$ , $\newline$ Cube root of $192$ = cube root of $(2^6 \times 3)$ = $2^2 \times$ cube root of $3$ = $4 \times$ cube root of $3$ .
Multiply results: Multiply the results. $\newline$ $(5 \times \sqrt[3]{3}) \times (4 \times \sqrt[3]{3}) = 20 \times (\sqrt[3]{3})^2$ .
Simplify further: Simplify further if possible. $\newline$ $(\sqrt[3]{3})^2$ is not further simplifiable without a calculator or more specific value approximation.

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