Simplifying Radical Expressions - Square Roots Properties

Simplify the following expression:
\[
\sqrt{\frac{9x^2}{y^3}}
\]

Solution
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:

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