Question

Simplify w/ positive values

Answer 1

$\bf \sqrt{\cfrac{3x^5}{75x^9}}\qquad \begin{cases} 75=3\cdot 5\cdot 5\to 3\cdot 5^2\\ x^9\to x^{5+4}\to x^5\cdot x^4 \end{cases}\qquad thus \\\\\\\\ \sqrt{\cfrac{3x^5}{3\cdot 5^2\cdot x^5\cdot x^4}}\implies \sqrt{\cfrac{3x^5}{3x^5}\cdot \cfrac{1}{5^2x^4}}\implies \sqrt{1\cdot \cfrac{1}{5^2x^4}} \\\\\\\\ \sqrt{\cfrac{1}{5^2x^4}}\qquad \begin{cases} 1^2\to 1\\ x^4\to x^{2\cdot 2}\to (x^2)^2 \end{cases}\qquad thus$

$\bf \sqrt{\cfrac{1}{5^2x^4}}\implies \sqrt{\cfrac{1^2}{5^2(x^2)^2}}\implies \sqrt{\cfrac{1^2}{(5x^2)^2}}\implies \sqrt{\left( \cfrac{1}{5x^2} \right)^2} \\\\\\ \cfrac{1}{5x^2}$