Simplify: \( \frac{\sqrt{25 x^{10} y^{8}}}{10 x^{2} y^{6}} \)

To simplify \( \frac{\sqrt{25 x^{10} y^{8}}}{10 x^{2} y^{6}} \), we first find the square root of the numerator: \[ \sqrt{25 x^{10} y^{8}} = \sqrt{25} \cdot \sqrt{x^{10}} \cdot \sqrt{y^{8}} = 5 \cdot x^{5} \cdot y^{4} = 5 x^{5} y^{4} \] Now we can rewrite the entire expression: \[ \frac{5 x^{5} y^{4}}{10 x^{2} y^{6}} \] Next, we simplify this fraction: \[ \frac{5}{10} = \frac{1}{2} \] Now for the variables, we subtract the exponents in the numerator and denominator: \[ x^{5} / x^{2} = x^{5-2} = x^{3} \quad \text{and} \quad y^{4} / y^{6} = y^{4-6} = y^{-2} \] Putting it all together, we have: \[ \frac{1}{2} x^{3} y^{-2} \] Since \(y^{-2}\) can be rewritten as \(\frac{1}{y^{2}}\), we finalize our expression as: \[ \frac{x^{3}}{2 y^{2}} \] Thus, the simplified form of \( \frac{\sqrt{25 x^{10} y^{8}}}{10 x^{2} y^{6}} \) is: \[ \frac{x^{3}}{2 y^{2}} \]