Simplify ( square root of a square root of b)/( fifth root of ab)

$\frac{\sqrt{a \sqrt{b}}}{\sqrt[5]{a b}}$

Multiply $\frac{\sqrt{a \sqrt{b}}}{\sqrt[5]{a b}}$ by $\frac{{\sqrt[5]{a b}}^{4}}{{\sqrt[5]{a b}}^{4}}$ .

$\frac{\sqrt{a \sqrt{b}}}{\sqrt[5]{a b}} \cdot \frac{{\sqrt[5]{a b}}^{4}}{{\sqrt[5]{a b}}^{4}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{a \sqrt{b}}}{\sqrt[5]{a b}}$ by $\frac{{\sqrt[5]{a b}}^{4}}{{\sqrt[5]{a b}}^{4}}$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[5]{a b}}^{4}}{\sqrt[5]{a b} {\sqrt[5]{a b}}^{4}}$

Raise $\sqrt[5]{a b}$ to the power of $1$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[5]{a b}}^{4}}{{\sqrt[5]{a b}}^{1} {\sqrt[5]{a b}}^{4}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[5]{a b}}^{4}}{{\sqrt[5]{a b}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[5]{a b}}^{4}}{{\sqrt[5]{a b}}^{5}}$

Rewrite ${\sqrt[5]{a b}}^{5}$ as $a b$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{a b}$ as ${(a b)}^{\frac{1}{5}}$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[5]{a b}}^{4}}{{({(a b)}^{\frac{1}{5}})}^{5}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[5]{a b}}^{4}}{{(a b)}^{\frac{1}{5} \cdot 5}}$

Combine $\frac{1}{5}$ and $5$ .

$\frac{\sqrt{a \sqrt{b}} {\sqrt[5]{a b}}^{4}}{{(a b)}^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[5]{a b}}^{4}}{{(a b)}^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[5]{a b}}^{4}}{{(a b)}^{1}}$

Simplify.

$\frac{\sqrt{a \sqrt{b}} {\sqrt[5]{a b}}^{4}}{a b}$

Simplify the numerator.

Rewrite ${\sqrt[5]{a b}}^{4}$ as ${({(a b)}^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt{a \sqrt{b}} \sqrt[5]{{(a b)}^{4}}}{a b}$

Apply the product rule to $a b$ .

$\frac{\sqrt{a \sqrt{b}} \sqrt[5]{a^{4} b^{4}}}{a b}$

Simplify the numerator.

Rewrite the expression using the least common index of $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a \sqrt{b}}$ as ${(a \sqrt{b})}^{\frac{1}{2}}$ .

$\frac{{(a \sqrt{b})}^{\frac{1}{2}} \sqrt[5]{a^{4} b^{4}}}{a b}$

Rewrite ${(a \sqrt{b})}^{\frac{1}{2}}$ as ${(a \sqrt{b})}^{\frac{5}{10}}$ .

$\frac{{(a \sqrt{b})}^{\frac{5}{10}} \sqrt[5]{a^{4} b^{4}}}{a b}$

Rewrite ${(a \sqrt{b})}^{\frac{5}{10}}$ as $\sqrt[10]{{(a \sqrt{b})}^{5}}$ .

$\frac{\sqrt[10]{{(a \sqrt{b})}^{5}} \sqrt[5]{a^{4} b^{4}}}{a b}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{a^{4} b^{4}}$ as ${(a^{4} b^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt[10]{{(a \sqrt{b})}^{5}} {(a^{4} b^{4})}^{\frac{1}{5}}}{a b}$

Rewrite ${(a^{4} b^{4})}^{\frac{1}{5}}$ as ${(a^{4} b^{4})}^{\frac{2}{10}}$ .

$\frac{\sqrt[10]{{(a \sqrt{b})}^{5}} {(a^{4} b^{4})}^{\frac{2}{10}}}{a b}$

Rewrite ${(a^{4} b^{4})}^{\frac{2}{10}}$ as $\sqrt[10]{{(a^{4} b^{4})}^{2}}$ .

$\frac{\sqrt[10]{{(a \sqrt{b})}^{5}} \sqrt[10]{{(a^{4} b^{4})}^{2}}}{a b}$

Combine using the product rule for radicals.

$\frac{\sqrt[10]{{(a \sqrt{b})}^{5} {(a^{4} b^{4})}^{2}}}{a b}$

Apply the product rule to $a \sqrt{b}$ .

$\frac{\sqrt[10]{a^{5} {\sqrt{b}}^{5} {(a^{4} b^{4})}^{2}}}{a b}$

Apply the product rule to $a^{4} b^{4}$ .

$\frac{\sqrt[10]{a^{5} {\sqrt{b}}^{5} {(a^{4})}^{2} {(b^{4})}^{2}}}{a b}$

Rewrite ${\sqrt{b}}^{5}$ as ${(b^{5})}^{\frac{1}{2}}$ .

$\frac{\sqrt[10]{a^{5} \sqrt{b^{5}} {(a^{4})}^{2} {(b^{4})}^{2}}}{a b}$

Rewrite $b^{5}$ as ${(b^{2})}^{2} b$ .

Factor out $b^{4}$ .

$\frac{\sqrt[10]{a^{5} \sqrt{b^{4} b} {(a^{4})}^{2} {(b^{4})}^{2}}}{a b}$

Rewrite $b^{4}$ as ${(b^{2})}^{2}$ .

$\frac{\sqrt[10]{a^{5} \sqrt{{(b^{2})}^{2} b} {(a^{4})}^{2} {(b^{4})}^{2}}}{a b}$

Pull terms out from under the radical.

$\frac{\sqrt[10]{a^{5} (b^{2} \sqrt{b}) {(a^{4})}^{2} {(b^{4})}^{2}}}{a b}$

Multiply the exponents in ${(a^{4})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt[10]{a^{5} (b^{2} \sqrt{b}) a^{4 \cdot 2} {(b^{4})}^{2}}}{a b}$

Multiply $4$ by $2$ .

$\frac{\sqrt[10]{a^{5} (b^{2} \sqrt{b}) a^{8} {(b^{4})}^{2}}}{a b}$

Multiply the exponents in ${(b^{4})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt[10]{a^{5} (b^{2} \sqrt{b}) a^{8} b^{4 \cdot 2}}}{a b}$

Multiply $4$ by $2$ .

$\frac{\sqrt[10]{a^{5} (b^{2} \sqrt{b}) a^{8} b^{8}}}{a b}$

$\frac{\sqrt[10]{a^{5} b^{2} \sqrt{b} a^{8} b^{8}}}{a b}$

Combine exponents.

Multiply $a^{5}$ by $a^{8}$ by adding the exponents.

Move $a^{8}$ .

$\frac{\sqrt[10]{a^{8} a^{5} b^{2} \sqrt{b} b^{8}}}{a b}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt[10]{a^{8 + 5} b^{2} \sqrt{b} b^{8}}}{a b}$

Add $8$ and $5$ .

$\frac{\sqrt[10]{a^{13} b^{2} \sqrt{b} b^{8}}}{a b}$

Multiply $b^{2}$ by $b^{8}$ by adding the exponents.

Move $b^{8}$ .

$\frac{\sqrt[10]{a^{13} (b^{8} b^{2}) \sqrt{b}}}{a b}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt[10]{a^{13} b^{8 + 2} \sqrt{b}}}{a b}$

Add $8$ and $2$ .

$\frac{\sqrt[10]{a^{13} b^{10} \sqrt{b}}}{a b}$

Simplify the numerator.

Rewrite $a^{13} b^{10} \sqrt{b}$ as ${(a b)}^{10} (a^{3} \sqrt{b})$ .

Factor out $a^{10}$ .

$\frac{\sqrt[10]{a^{10} a^{3} b^{10} \sqrt{b}}}{a b}$

Move $a^{3}$ .

$\frac{\sqrt[10]{a^{10} b^{10} a^{3} \sqrt{b}}}{a b}$

Rewrite $a^{10} b^{10}$ as ${(a b)}^{10}$ .

$\frac{\sqrt[10]{{(a b)}^{10} a^{3} \sqrt{b}}}{a b}$

Add parentheses.

$\frac{\sqrt[10]{{(a b)}^{10} (a^{3} \sqrt{b})}}{a b}$

Pull terms out from under the radical.

$\frac{a b \sqrt[10]{a^{3} \sqrt{b}}}{a b}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $a$ .

Cancel the common factor.

$\frac{a b \sqrt[10]{a^{3} \sqrt{b}}}{a b}$

Rewrite the expression.

$\frac{b \sqrt[10]{a^{3} \sqrt{b}}}{b}$

Cancel the common factor of $b$ .

Cancel the common factor.

$\frac{b \sqrt[10]{a^{3} \sqrt{b}}}{b}$

Divide $\sqrt[10]{a^{3} \sqrt{b}}$ by $1$ .

$\sqrt[10]{a^{3} \sqrt{b}}$

Simplify ( square root of a square root of b)/( fifth root of ab)

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