a) \(ab\sqrt{\dfrac{a}{b}}=ab\sqrt{\dfrac{a\cdot b}{b\cdot b}}=ab\sqrt{\dfrac{ab}{b^2}}=\dfrac{ab\sqrt{ab}}{\sqrt{b^2}}=\dfrac{ab\sqrt{ab}}{\left|b\right|}\)
b) \(\dfrac{a}{b}\sqrt{\dfrac{b}{a}}=\dfrac{a}{b}\sqrt{\dfrac{b\cdot a}{a\cdot a}}=\dfrac{a}{b}\sqrt{\dfrac{ab}{a^2}}=\dfrac{a}{b}\dfrac{\sqrt{ab}}{\sqrt{a^2}}=\dfrac{a\sqrt{ab}}{b\left|a\right|}\)
c) \(\sqrt{\dfrac{1}{b}+\dfrac{1}{b^2}}=\sqrt{\dfrac{b}{b^2}+\dfrac{1}{b^2}}=\sqrt{\dfrac{b+1}{b^2}}=\dfrac{\sqrt{b+1}}{\sqrt{b^2}}=\dfrac{\sqrt{b+1}}{\left|b\right|}\)
d) \(\sqrt{\dfrac{9a^3}{36b}}=\sqrt{\dfrac{9a^3\cdot b}{36b\cdot b}}=\sqrt{\dfrac{9a^3b}{36b^2}}=\dfrac{\sqrt{9a^3b}}{\sqrt{\left(6b\right)^2}}=\dfrac{3\left|a\right|\sqrt{b}}{6\left|b\right|}=\dfrac{\left|a\right|\sqrt{b}}{\left|b\right|}\)
e) \(3xy\sqrt{\dfrac{2}{xy}}=3xy\sqrt{\dfrac{2\cdot xy}{xy\cdot xy}}=3xy\sqrt{\dfrac{2xy}{\left(xy\right)^2}}=3xy\dfrac{\sqrt{2xy}}{\sqrt{\left(xy\right)^2}}=\dfrac{3xy\sqrt{2xy}}{\left|xy\right|}\)
