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Compute the multiplication and reduce to the simplest expression. $\newline$ $\sqrt{32} \times \sqrt{2}$

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Multiply radical expressions

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Q. Compute the multiplication and reduce to the simplest expression.

\newline

\sqrt{32} \times \sqrt{2}

Factorization of $\sqrt{32}$ and $\sqrt{2}$ : Find the prime factorization of $\sqrt{32}$ and $\sqrt{2}$ . The prime factorization of $32$ is $2^5$ , and the prime factorization of $2$ is just $2$ . So, $\sqrt{32} \times \sqrt{2}$ can be expressed as $\sqrt{2^5} \times \sqrt{2}$ .
Combine square roots: Apply the multiplication property of square roots to combine them. So, $\sqrt{2^5} \times \sqrt{2} = \sqrt{2^5 \times 2}$ .
Combine exponents: Combine the exponents of $2$ inside the square root. So, $\sqrt{2^5 \times 2} = \sqrt{2^6}$ .
Simplify square root: Simplify $\sqrt{2^6}$ by taking the square root of $2^6$ . Since $2^6$ is a perfect square $(2^3)^2$ , $\sqrt{2^6} = 2^3$ .
Calculate final answer: Calculate $2^3$ to get the final answer. So, $2^3 = 8$ .

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