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Simplify. Assume $q$ is greater than or equal to zero. $\newline$ $\sqrt{8q^2}$

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Simplify radical expressions with variables

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Q. Simplify. Assume

q

is greater than or equal to zero.

\newline

\sqrt{8q^2}

Factorize $8q^2$ : Factor $8q^2$ into its prime factors and pair the squares. $\newline$ The prime factorization of $8$ is $2 \times 2 \times 2$ , and $q^2$ is already a square. So, we have: $\newline$ $\sqrt{8q^2} = \sqrt{2 \times 2 \times 2 \times q \times q}$
Group Factors into Pairs: Group the factors into pairs of squares. $\newline$ We can group the factors as follows: $\newline$ $\sqrt{2 \times 2 \times 2 \times q \times q} = \sqrt{(2 \times 2) \times (q \times q) \times 2}$ $\newline$ = $\sqrt{4 \times q^2 \times 2}$
Simplify Square Roots: Simplify the square root of the squares. $\newline$ Since the square root of a square is just the base, we can simplify: $\newline$ $\sqrt{4 \cdot q^2 \cdot 2} = \sqrt{4} \cdot \sqrt{q^2} \cdot \sqrt{2}$ $\newline$ $= 2 \cdot q \cdot \sqrt{2}$

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