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Simplify. Assume $x$ is greater than or equal to zero. $\newline$ $\sqrt{75x^9}$

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Simplify radical expressions with variables

Full solution

Q. Simplify. Assume

x

is greater than or equal to zero.

\newline

\sqrt{75x^9}

Factorization: Factor $75x^9$ into its prime factors and perfect squares. $\newline$ $75$ can be factored into $3 \times 5 \times 5$ , and $x^9$ is $(x^4)^2 \times x$ . So, we have: $\newline$ $\sqrt{75x^9} = \sqrt{3 \times 5^2 \times (x^4)^2 \times x}$
Separation: Separate the perfect squares from the non-perfect squares inside the radical. $\newline$ We can rewrite the expression as: $\newline$ $\sqrt{5^2 \cdot (x^4)^2 \cdot 3x}$
Square Roots: Take the square root of the perfect squares. $\newline$ The square root of $5^2$ is $5$ , and the square root of $(x^4)^2$ is $x^4$ . We then have: $\newline$ $5 \times x^4 \times \sqrt{3x}$
Final Expression: Write the final simplified expression. $\newline$ The simplified form of the original expression is: $\newline$ $5x^4 \cdot \sqrt{3x}$

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