Simplify. Assume d is greater than or equal to zero. sqrt 45d^10 ______

Factor and Express: sqrt(45d^10) First, let's factor the radicand (the number inside the square root) into its prime factors and express the variable part with exponents. sqrt(45d^10) = sqrt(3 * 3 * 5 * d^10)Group Identical Factors: sqrt(3 * 3 * 5 * d^10) Now, group the identical factors and express the exponents for the variable part. sqrt(3 * 3 * 5 * d^10) = sqrt(3^2 * 5 * d^(2*5))Apply Product Property: sqrt(3^2 * 5 * d^(2*5)) Apply the product property of radicals, which allows us to take the square root of each factor separately. sqrt(3^2 * 5 * d^(2*5)) = sqrt(3^2) * sqrt(5) * sqrt(d^(2*5))Simplify Square Roots: sqrt(3^2) * sqrt(5) * sqrt(d^(2*5)) Simplify the square roots where possible. The square root of a square number is just the base of the square, and the square root of a variable raised to an even power is the variable raised to half that power. sqrt(3^2) * sqrt(5) * sqrt(d^(2*5)) = 3 * sqrt(5) * d^5

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Simplify. Assume $d$ is greater than or equal to zero. $\newline$ $\sqrt{45d^{10}}$

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Q. Simplify. Assume

d

is greater than or equal to zero.

\newline

\sqrt{45d^{10}}

Factor and Express: $\sqrt{45d^{10}}$ $\newline$ First, let's factor the radicand (the number inside the square root) into its prime factors and express the variable part with exponents. $\newline$ $\sqrt{45d^{10}} = \sqrt{3 \times 3 \times 5 \times d^{10}}$
Group Identical Factors: $\sqrt{3 \times 3 \times 5 \times d^{10}}$ $\newline$ Now, group the identical factors and express the exponents for the variable part. $\newline$ $\sqrt{3 \times 3 \times 5 \times d^{10}} = \sqrt{3^2 \times 5 \times d^{2\times5}}$
Apply Product Property: $\sqrt{3^2 \times 5 \times d^{2\times5}}$ $\newline$ Apply the product property of radicals, which allows us to take the square root of each factor separately. $\newline$ $\sqrt{3^2 \times 5 \times d^{2\times5}} = \sqrt{3^2} \times \sqrt{5} \times \sqrt{d^{2\times5}}$
Simplify Square Roots: $\sqrt{3^2} \times \sqrt{5} \times \sqrt{d^{2\times5}}$ Simplify the square roots where possible. The square root of a square number is just the base of the square, and the square root of a variable raised to an even power is the variable raised to half that power. $\sqrt{3^2} \times \sqrt{5} \times \sqrt{d^{2\times5}} = 3 \times \sqrt{5} \times d^5$