Answer
$\sqrt [4] {a^3b^3 }$.
Work Step by Step
The given expression is
$\frac{\sqrt[4] {a^5b^5}}{\sqrt {ab}}$
Use $\sqrt[n] x =x^{\frac{1}{n}}$.
$=\frac {\left ( a^5b^5 \right )^{\frac{1}{4}} }{\left ( ab \right )^{\frac{1}{2}}}$
Use $\frac{1}{x^{n}}=x^{-n}$.
$=\left ( a^5b^5 \right )^{\frac{1}{4}} \cdot \left ( ab \right )^{-\frac{1}{2}}$
Use $(ab)^m=a^mb^m$.
$=a^{\frac{5}{4}}b^{\frac{5}{4}} \cdot a^{-\frac{1}{2}}b^{-\frac{1}{2}}$
Use $a^m\cdot a^n=a^{m+n}$.
$=a^{\frac{5}{4}-\frac{1}{2}}b^{\frac{5}{4}-\frac{1}{2}} $
$=a^{\frac{5-2}{4}}b^{\frac{5-2}{4}} $
$=a^{\frac{3}{4}}b^{\frac{3}{4}} $
Use $a^mb^m=(ab)^m$.
$=\left ( a^3b^3 \right )^{\frac{1}{4}}$
Use $x^{\frac{1}{n}}=\sqrt[n] x $.
$=\sqrt [4] {a^3b^3 }$.
