Simplify. Assume f is greater than or equal to zero. sqrt 12f^7 ______

Factor and Express: sqrt(12f^7) First, we need to factor the radicand (the number inside the square root) into its prime factors and express the variable f in a way that will allow us to simplify the square root. sqrt(12f^7) = sqrt(2 * 2 * 3 * f^7)Group and Simplify: sqrt(2 * 2 * 3 * f^7) Now, we group the identical factors and express f^7 as f^6 * f to make use of the even exponent. sqrt(2 * 2 * 3 * f^6 * f) = sqrt(2^2 * 3 * f^6 * f)Apply Product Property: sqrt(2^2 * 3 * f^6 * f) We apply the product property of radicals, which allows us to take the square root of each factor separately. sqrt(2^2 * 3 * f^6 * f) = sqrt(2^2) * sqrt(3) * sqrt(f^6) * sqrt(f)Simplify Perfect Squares: sqrt(12f^7) = sqrt(2^2) * sqrt(3) * sqrt(f^6) * sqrt(f) We simplify the square roots of the perfect squares, which are 2^2 and f^6. sqrt(2^2) * sqrt(3) * sqrt(f^6) * sqrt(f) = 2 * sqrt(3) * f^3 * sqrt(f)Final Simplification: Final Simplification Combine the constants and the variables outside the square root to get the final simplified form. 2 * f^3 * sqrt(3) * sqrt(f) = 2f^3 * sqrt(3f)

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Simplify. Assume $f$ is greater than or equal to zero. $\newline$ $\sqrt{12f^7}$

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Algebra 1

Simplify radical expressions: mixed review

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Q. Simplify. Assume

f

is greater than or equal to zero.

\newline

\sqrt{12f^7}

Factor and Express: $\sqrt{12f^7}$ $\newline$ First, we need to factor the radicand (the number inside the square root) into its prime factors and express the variable $f$ in a way that will allow us to simplify the square root. $\newline$ $\sqrt{12f^7} = \sqrt{2 \times 2 \times 3 \times f^7}$
Group and Simplify: $\sqrt{2 \times 2 \times 3 \times f^7}$ Now, we group the identical factors and express $f^7$ as $f^6 \times f$ to make use of the even exponent. $\sqrt{2 \times 2 \times 3 \times f^6 \times f} = \sqrt{2^2 \times 3 \times f^6 \times f}$
Apply Product Property: $\sqrt{2^2 \times 3 \times f^6 \times f}$ $\newline$ We apply the product property of radicals, which allows us to take the square root of each factor separately. $\newline$ $\sqrt{2^2 \times 3 \times f^6 \times f} = \sqrt{2^2} \times \sqrt{3} \times \sqrt{f^6} \times \sqrt{f}$
Simplify Perfect Squares: $\sqrt{12f^7} = \sqrt{2^2} \cdot \sqrt{3} \cdot \sqrt{f^6} \cdot \sqrt{f}$ We simplify the square roots of the perfect squares, which are $2^2$ and $f^6$ . $\sqrt{2^2} \cdot \sqrt{3} \cdot \sqrt{f^6} \cdot \sqrt{f} = 2 \cdot \sqrt{3} \cdot f^3 \cdot \sqrt{f}$
Final Simplification: Final Simplification $\newline$ Combine the constants and the variables outside the square root to get the final simplified form. $\newline$ $2 \cdot f^3 \cdot \sqrt{3} \cdot \sqrt{f} = 2f^3 \cdot \sqrt{3f}$