Simplify. Assume n is greater than or equal to zero. sqrt 75n^7 ______

Factorization and Pairing: Factor 75n^7 into its prime factors and pair the factors to form perfect squares where possible. 75 can be factored into 3 * 5 * 5, and n^7 can be written as n^6 * n, where n^6 is a perfect square since it has an even exponent. So, we have sqrt(75n^7) = sqrt(3 * 5^2 * n^6 * n).Separation of Squares: Separate the perfect squares from the non-perfect squares inside the square root. We can rewrite the expression as sqrt(5^2 * n^6) * sqrt(3n).Simplification: Simplify the square root of the perfect squares. Since sqrt(5^2) is 5 and sqrt(n^6) is n^3 (because the square root of n^6 is the number n raised to the power of 6/2, which is 3), we get: 5 * n^3 * sqrt(3n).Final Expression: Write the final simplified expression. The final simplified expression is 5n^3 * sqrt(3n).

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Simplify. Assume $n$ is greater than or equal to zero. $\newline$ $\sqrt{75n^7}$

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Algebra 1

Simplify radical expressions with variables

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Q. Simplify. Assume

n

is greater than or equal to zero.

\newline

\sqrt{75n^7}

Factorization and Pairing: Factor $75n^7$ into its prime factors and pair the factors to form perfect squares where possible. $\newline$ $75$ can be factored into $3 \times 5 \times 5$ , and $n^7$ can be written as $n^6 \times n$ , where $n^6$ is a perfect square since it has an even exponent. $\newline$ So, we have $\sqrt{75n^7} = \sqrt{3 \times 5^2 \times n^6 \times n}$ .
Separation of Squares: Separate the perfect squares from the non-perfect squares inside the square root. We can rewrite the expression as $\sqrt{5^2 \times n^6} \times \sqrt{3n}$ .
Simplification: Simplify the square root of the perfect squares. $\newline$ Since $\sqrt{5^2}$ is $5$ and $\sqrt{n^6}$ is $n^3$ (because the square root of $n^6$ is the number $n$ raised to the power of $\frac{6}{2}$ , which is $3$ ), we get: $\newline$ $5 \times n^3 \times \sqrt{3n}$ .
Final Expression: Write the final simplified expression. $\newline$ The final simplified expression is $5n^3 \cdot \sqrt{3n}$ .