Simplify. Assume f is greater than or equal to zero. sqrt 18f^8 ______

Identify Factorization: Identify the complete factorization of 18f^8. Complete factorization of 18f^8 is 2 * 3 * 3 * f^8 since 18 can be factored into 2 and 3^2, and f^8 is already a power of a prime number.Rewrite as Product: Rewrite sqrt(18f^8) as sqrt(2 * 3^2 * f^8). We can now express the square root of the product as the product of the square roots of the factors, keeping in mind that we can only take the square root of perfect squares directly.Simplify Perfect Squares: Simplify the square root of the perfect squares. Since 3^2 is a perfect square and f^8 is a perfect square (because 8 is an even exponent), we can take the square root of these directly. sqrt(18f^8) = sqrt(2) * sqrt(3^2) * sqrt(f^8)Calculate Square Roots: Calculate the square roots of the perfect squares. sqrt(3^2) is 3, and sqrt(f^8) is f^4 (since the square root of f to an even power is f to half that power). So we have sqrt(18f^8) = sqrt(2) * 3 * f^4Write Final Form: Write the final simplified form. The final simplified form is 3f^4 * sqrt(2), since sqrt(2) cannot be simplified further as it is not a perfect square.

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Simplify. Assume $f$ is greater than or equal to zero. $\newline$ $\sqrt{18f^8}$

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Algebra 1

Simplify radical expressions with variables

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Q. Simplify. Assume

f

is greater than or equal to zero.

\newline

\sqrt{18f^8}

Identify Factorization: Identify the complete factorization of $18f^8$ . Complete factorization of $18f^8$ is $2 \times 3 \times 3 \times f^8$ since $18$ can be factored into $2$ and $3^2$ , and $f^8$ is already a power of a prime number.
Rewrite as Product: Rewrite $\sqrt{18f^8}$ as $\sqrt{2 \times 3^2 \times f^8}$ . We can now express the square root of the product as the product of the square roots of the factors, keeping in mind that we can only take the square root of perfect squares directly.
Simplify Perfect Squares: Simplify the square root of the perfect squares. Since $3^2$ is a perfect square and $f^8$ is a perfect square (because $8$ is an even exponent), we can take the square root of these directly. $\sqrt{18f^8} = \sqrt{2} \times \sqrt{3^2} \times \sqrt{f^8}$
Calculate Square Roots: Calculate the square roots of the perfect squares. $\sqrt{3^2}$ is $3$ , and $\sqrt{f^8}$ is $f^4$ (since the square root of $f$ to an even power is $f$ to half that power). So we have $\sqrt{18f^8} = \sqrt{2} \times 3 \times f^4$
Write Final Form: Write the final simplified form. $\newline$ The final simplified form is $3f^4 \times \sqrt{2}$ , since $\sqrt{2}$ cannot be simplified further as it is not a perfect square.