[tex]\bf \sqrt{\cfrac{3x^5}{75x^9}}\qquad
\begin{cases}
75=3\cdot 5\cdot 5\to 3\cdot 5^2\\
x^9\to x^{5+4}\to x^5\cdot x^4
\end{cases}\qquad thus
\\\\\\\\
\sqrt{\cfrac{3x^5}{3\cdot 5^2\cdot x^5\cdot x^4}}\implies \sqrt{\cfrac{3x^5}{3x^5}\cdot \cfrac{1}{5^2x^4}}\implies \sqrt{1\cdot \cfrac{1}{5^2x^4}}
\\\\\\\\
\sqrt{\cfrac{1}{5^2x^4}}\qquad
\begin{cases}
1^2\to 1\\
x^4\to x^{2\cdot 2}\to (x^2)^2
\end{cases}\qquad thus
\\\\
[/tex]
[tex]\bf \sqrt{\cfrac{1}{5^2x^4}}\implies \sqrt{\cfrac{1^2}{5^2(x^2)^2}}\implies \sqrt{\cfrac{1^2}{(5x^2)^2}}\implies \sqrt{\left( \cfrac{1}{5x^2} \right)^2}
\\\\\\
\cfrac{1}{5x^2}[/tex]
