Simplify. Assume m is greater than or equal to zero. sqrt 27m^4 ______

Factor and Express: sqrt(27m^4) First, let's factor the radicand into its prime factors and express m^4 as a square of squares. sqrt(27m^4) = sqrt(3 * 3 * 3 * m^2 * m^2)Group and Simplify: sqrt(3 * 3 * 3 * m^2 * m^2) Now, group the identical factors and use the exponents to simplify the expression. sqrt(3 * 3 * 3 * m^2 * m^2) = sqrt(3^3 * (m^2)^2)Apply Product Property: sqrt(3^3 * (m^2)^2) Apply the product property of radicals, which states that sqrt(a * b) = sqrt(a) * sqrt(b). sqrt(3^3 * (m^2)^2) = sqrt(3^2) * sqrt(3) * sqrt((m^2)^2)Simplify Square Roots: sqrt(3^2) * sqrt(3) * sqrt((m^2)^2) Simplify the square roots where possible. Since sqrt(3^2) and sqrt((m^2)^2) are perfect squares, they simplify to 3 and m^2, respectively. sqrt(3^2) * sqrt(3) * sqrt((m^2)^2) = 3 * sqrt(3) * m^2Combine Constants and Variables: 3 * sqrt(3) * m^2 Combine the constants and variables outside the square root to get the final simplified form. 3 * sqrt(3) * m^2 = 3m^2 * sqrt(3)

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Simplify. Assume $m$ is greater than or equal to zero. $\newline$ $\sqrt{27m^4}$

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Q. Simplify. Assume

m

is greater than or equal to zero.

\newline

\sqrt{27m^4}

Factor and Express: $\sqrt{27m^4}$ $\newline$ First, let's factor the radicand into its prime factors and express $m^4$ as a square of squares. $\newline$ $\sqrt{27m^4} = \sqrt{3 \times 3 \times 3 \times m^2 \times m^2}$
Group and Simplify: $\sqrt{3 \times 3 \times 3 \times m^2 \times m^2}$ $\newline$ Now, group the identical factors and use the exponents to simplify the expression. $\newline$ $\sqrt{3 \times 3 \times 3 \times m^2 \times m^2} = \sqrt{3^3 \times (m^2)^2}$
Apply Product Property: $\sqrt{3^3 \cdot (m^2)^2}$ $\newline$ Apply the product property of radicals, which states that $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ . $\newline$ $\sqrt{3^3 \cdot (m^2)^2} = \sqrt{3^2} \cdot \sqrt{3} \cdot \sqrt{(m^2)^2}$
Simplify Square Roots: $\sqrt{3^2} \times \sqrt{3} \times \sqrt{(m^2)^2}$ Simplify the square roots where possible. Since $\sqrt{3^2}$ and $\sqrt{(m^2)^2}$ are perfect squares, they simplify to $3$ and $m^2$ , respectively. $\sqrt{3^2} \times \sqrt{3} \times \sqrt{(m^2)^2} = 3 \times \sqrt{3} \times m^2$
Combine Constants and Variables: $3 \times \sqrt{3} \times m^2$ $\newline$ Combine the constants and variables outside the square root to get the final simplified form. $\newline$ $3 \times \sqrt{3} \times m^2 = 3m^2 \times \sqrt{3}$