Question

which expression is equivalent to \\( \sqrt{\frac{225}{625} m^4 n^8} \\)?
\\( \frac{1}{20} m^2 (n^3) \\)
\\( \frac{1}{20} m^2 n^4 \\)
\\( \frac{3}{5} m^2 (n^3) \\)
\\( \frac{3}{5} m^2 n^4 \\)

Explanation:

Step1: Simplify the square root of the fraction

We know that $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$, so $\sqrt{\frac{225}{625}m^{4}n^{8}}=\frac{\sqrt{225m^{4}n^{8}}}{\sqrt{625}}$.

Step2: Simplify the square roots of numerator and denominator

First, simplify $\sqrt{225}=15$, $\sqrt{625}=25$, $\sqrt{m^{4}} = m^{2}$ (because $(m^{2})^{2}=m^{4}$), and $\sqrt{n^{8}}=n^{4}$ (because $(n^{4})^{2}=n^{8}$). So the numerator becomes $\sqrt{225m^{4}n^{8}} = 15m^{2}n^{4}$ and the denominator is $25$.

Step3: Simplify the fraction

Now, simplify $\frac{15m^{2}n^{4}}{25}$. We can divide 15 and 25 by their greatest common divisor, which is 5. So $\frac{15\div5}{25\div5}=\frac{3}{5}$, and we keep $m^{2}n^{4}$. So the expression simplifies to $\frac{3}{5}m^{2}n^{4}$.

Answer:

$\frac{3}{5}m^{2}n^{4}$ (the last option, assuming the last option is $\boldsymbol{\frac{3}{5}m^{2}n^{4}}$)