Simplifying Radical Expressions – Square Roots Properties

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We are tasked with simplifying and solving the following expression:
\[
\sqrt{\frac{9x^2}{y^3}}
\]

Step-by-Step Simplification:

Separate the square root of a fraction: The square root of a fraction can be written as the square root of the numerator divided by the square root of the denominator:


\[
\sqrt{\frac{9x^2}{y^3}} = \frac{\sqrt{9x^2}}{\sqrt{y^3}}
\]

Simplify the numerator: The square root of \(9x^2\) can be simplified:


\[
\sqrt{9x^2} = \sqrt{9} \cdot \sqrt{x^2} = 3x
\]

Simplify the denominator: To simplify \(\sqrt{y^3}\), we can break it down:


\[
\sqrt{y^3} = \sqrt{y^2 \cdot y} = \sqrt{y^2} \cdot \sqrt{y} = y\sqrt{y}
\]

Put everything together: Now, we can rewrite the simplified expression as:


\[
\frac{3x}{y\sqrt{y}}
\]

Final Simplified Expression:

\(\frac{3x}{y\sqrt{y}}\) is the simplified form of \(\sqrt{\frac{9x^2}{y^3}}\)

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