Question

\sqrt{\frac{25x^{6}y^{3}}{16x^{2}y^{9}}}

Explanation:

Step1: Simplify the fraction inside the square root

First, simplify the coefficients and the variables separately. For the coefficients, we have $\frac{25}{16}$. For the $x$ terms, using the rule of exponents $a^m\div a^n = a^{m - n}$, we get $x^{6-2}=x^4$. For the $y$ terms, we have $y^{3 - 9}=y^{-6}$. So the expression inside the square root becomes $\frac{25x^4y^{-6}}{16}$.

Step2: Take the square root of each part

The square root of $\frac{25}{16}$ is $\frac{5}{4}$. The square root of $x^4$ is $x^{4\div2}=x^2$. The square root of $y^{-6}$ is $y^{-6\div2}=y^{-3}$, which can be written as $\frac{1}{y^3}$ using the rule $a^{-n}=\frac{1}{a^n}$.

Step3: Combine the parts

Multiplying these together, we get $\frac{5x^2}{4y^3}$.

Answer:

$\frac{5x^2}{4y^3}$