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Simplify.
Remove all perfect squares from inside the square root. Assume
a is positive.

sqrt(108a^(6))=

Simplify. $\newline$ Remove all perfect squares from inside the square root. Assume $\newline$ $a$ is positive. $\newline$ $\sqrt{108a^{6}}$

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Simplify radical expressions with variables

Full solution

Q. Simplify.

\newline

Remove all perfect squares from inside the square root. Assume

\newline

a

is positive.

\newline

\sqrt{108a^{6}}

Factorize and Identify Perfect Squares: To simplify $\sqrt{108a^{6}}$ , we first factor $108$ into its prime factors and express $a^{6}$ as $(a^{3})^{2}$ to identify perfect squares. $\newline$ $108$ can be factored as $2 \times 54$ , which further factors to $2 \times 2 \times 27$ , and finally to $2 \times 2 \times 3 \times 9$ . Since $9$ is a perfect square $(3^{2})$ , we can take it out of the square root. Also, $a^{6}$ is a perfect square since it can be written as $(a^{3})^{2}$ . $\newline$ So, $108$ $1$ .
Take Out Perfect Squares: Now we take the square root of the perfect squares and place them outside the square root. The perfect squares are $3^2$ and $(a^{3})^2$ . So, $\sqrt{108a^{6}} = \sqrt{2 \times 2 \times 3 \times 3^2 \times (a^{3})^2} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{(a^{3})^2} \times \sqrt{3}$ . This simplifies to $2 \times 3 \times a^{3} \times \sqrt{3}$ .
Combine and Simplify: Finally, we combine the numbers and variables outside the square root to get the simplified expression. $\newline$ So, $\sqrt{108a^{6}} = 2 \times 3 \times a^{3} \times \sqrt{3} = 6a^{3} \times \sqrt{3}.$

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