To simplify the expression [tex]\(\sqrt{\frac{5 x^2}{9 y^2}}\)[/tex], follow these steps:
1. Rewrite the Expression Inside the Square Root:
The given expression is [tex]\(\sqrt{\frac{5 x^2}{9 y^2}}\)[/tex].
2. Separate the Fraction:
Break this down into two separate square roots:
[tex]\[
\sqrt{\frac{5 x^2}{9 y^2}} = \sqrt{\frac{5}{9}} \cdot \sqrt{\frac{x^2}{y^2}}
\][/tex]
3. Simplify Each Square Root Separately:
- For [tex]\(\sqrt{\frac{5}{9}}\)[/tex]:
[tex]\[
\sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{\sqrt{9}} = \frac{\sqrt{5}}{3}
\][/tex]
- For [tex]\(\sqrt{\frac{x^2}{y^2}}\)[/tex]:
[tex]\[
\sqrt{\frac{x^2}{y^2}} = \frac{\sqrt{x^2}}{\sqrt{y^2}} = \frac{x}{y} \quad (\text{assuming both } x \text{ and } y \text{ are positive})
\][/tex]
4. Combine the Results:
Now, multiply the simplified results of the square roots together:
[tex]\[
\sqrt{\frac{5 x^2}{9 y^2}} = \frac{\sqrt{5}}{3} \cdot \frac{x}{y} = \frac{\sqrt{5} \cdot x}{3 \cdot y}
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
\sqrt{\frac{5 x^2}{9 y^2}} = \frac{\sqrt{5} \cdot x}{3 \cdot y}
\][/tex]
