Simplify:

$$ \frac{\sqrt{a\sqrt{b}}}{\sqrt{ab}} $$

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Rewrite terms with fractional exponents.

$$ \frac{\sqrt{a\sqrt{b}}}{\sqrt{ab}} = \frac{(ab^{\frac{1}{2}})^{\frac{1}{2}}}{(ab)^{\frac{1}{2}}} $$

Using a few exponent rules,

$$ \frac{(ab^{\frac{1}{2}})^{\frac{1}{2}}}{(ab)^{\frac{1}{2}}} = \frac{a^{\frac{1}{2}}b^{\frac{1}{2}\cdot\frac{1}{2}}}{a^{\frac{1}{2}}b^{\frac{1}{2}}}$$ $$ = \frac{\cancel{a^{\frac{1}{2}}}b^{\frac{1}{4}}}{\cancel{a^{\frac{1}{2}}}b^{\frac{1}{2}}} = \frac{b^{\frac{1}{4}}}{b^{\frac{1}{2}}} $$ $$ = b^{\frac{1}{4}-\frac{1}{2}} = b^{-\frac{1}{4}}$$ $$ = \boxed{\frac{1}{\sqrt[4]{b}}} $$

Looking at the individual factors underneath the square roots:

$$ \frac{\sqrt{a\sqrt{b}}}{\sqrt{ab}} = \frac{\cancel{\sqrt{a}}\cdot\sqrt{\sqrt{b}}}{\cancel{\sqrt{a}}\cdot\sqrt{b}} $$ $$ = \frac{\sqrt{\sqrt{b}}}{\sqrt{b}} $$

Like the first approach, the next part is much easier to simplify with fractional exponents.

$$ = \frac{\sqrt{\sqrt{b}}}{\sqrt{b}} = \frac{(b^\frac{1}{2})^{\frac{1}{2}}}{b^{\frac{1}{2}}}$$ $$ = \frac{b^{\frac{1}{4}}}{b^{\frac{1}{2}} } $$ $$ = b^{\frac{1}{4}-\frac{1}{2}} = b^{-\frac{1}{4}}$$ $$ = \boxed{\frac{1}{\sqrt[4]{b}}} $$