Answer
$\frac{x^{18}}{4y^{22}}$
Work Step by Step
Based on the power of a product rule, we know that $(ab)^{n}=a^{n}b^{n}$ (where $n$ is a positive integer and $a$ and $b$ are real numbers).
Therefore, $(\frac{5x^{9}}{10y^{11}})^{2}=\frac{5^{2}(x^{9})^{2}}{10^{2}(y^{11})^{2}}=\frac{25(x^{9})^{2}}{100(y^{11})^{2}}=\frac{(x^{9})^{2}}{4(y^{11})^{2}}$.
Based on the power rule for exponents, we know that $(a^{m})^{n}=a^{mn}$ (where $m$ and $n$ are positive integers and $a$ is a real number).
Therefore, $\frac{(x^{9})^{2}}{4(y^{11})^{2}}=\frac{x^{9\times2}}{4y^{11\times2}}=\frac{x^{18}}{4y^{22}}$.
