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Simplify. Assume $x$ is greater than or equal to zero. $\newline$ $\sqrt{45x^5}$

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Simplify radical expressions with variables

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Q. Simplify. Assume

x

is greater than or equal to zero.

\newline

\sqrt{45x^5}

Factorize and Pair Factors: Factor $45x^5$ into its prime factors and pair the factors to simplify the square root. $\newline$ The prime factorization of $45$ is $3 \times 3 \times 5$ , and $x^5$ can be written as $x^2 \times x^2 \times x$ . So, we have: $\newline$ $\sqrt{45x^5} = \sqrt{3 \times 3 \times 5 \times x^2 \times x^2 \times x}$
Identify Perfect Squares: Identify the perfect squares inside the radical and rewrite the expression. $\newline$ We can pair the factors as follows: $\newline$ $\sqrt{45x^5} = \sqrt{(3^2) \cdot (x^2) \cdot (x^2) \cdot 5 \cdot x}$
Simplify Square Root: Simplify the square root by taking out the square factors. $\newline$ Since the square root of a square is just the base, we can simplify as follows: $\newline$ $\sqrt{45x^5} = \sqrt{3^2} \cdot \sqrt{x^2} \cdot \sqrt{x^2} \cdot \sqrt{5} \cdot \sqrt{x}$ $\newline$ $= 3 \cdot x \cdot x \cdot \sqrt{5} \cdot \sqrt{x}$ $\newline$ $= 3x^2 \cdot \sqrt{5x}$

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