Simplify ( square root of x square root of y)/( fifth root of x^2y)

$\frac{\sqrt{x \sqrt{y}}}{\sqrt[5]{x^{2} y}}$

Multiply $\frac{\sqrt{x \sqrt{y}}}{\sqrt[5]{x^{2} y}}$ by $\frac{{\sqrt[5]{x^{2} y}}^{4}}{{\sqrt[5]{x^{2} y}}^{4}}$ .

$\frac{\sqrt{x \sqrt{y}}}{\sqrt[5]{x^{2} y}} \cdot \frac{{\sqrt[5]{x^{2} y}}^{4}}{{\sqrt[5]{x^{2} y}}^{4}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{x \sqrt{y}}}{\sqrt[5]{x^{2} y}}$ by $\frac{{\sqrt[5]{x^{2} y}}^{4}}{{\sqrt[5]{x^{2} y}}^{4}}$ .

$\frac{\sqrt{x \sqrt{y}} {\sqrt[5]{x^{2} y}}^{4}}{\sqrt[5]{x^{2} y} {\sqrt[5]{x^{2} y}}^{4}}$

Raise $\sqrt[5]{x^{2} y}$ to the power of $1$ .

$\frac{\sqrt{x \sqrt{y}} {\sqrt[5]{x^{2} y}}^{4}}{{\sqrt[5]{x^{2} y}}^{1} {\sqrt[5]{x^{2} y}}^{4}}$

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

$\frac{\sqrt{x \sqrt{y}} {\sqrt[5]{x^{2} y}}^{4}}{{\sqrt[5]{x^{2} y}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{\sqrt{x \sqrt{y}} {\sqrt[5]{x^{2} y}}^{4}}{{\sqrt[5]{x^{2} y}}^{5}}$

Rewrite ${\sqrt[5]{x^{2} y}}^{5}$ as $x^{2} y$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2} y}$ as ${(x^{2} y)}^{\frac{1}{5}}$ .

$\frac{\sqrt{x \sqrt{y}} {\sqrt[5]{x^{2} y}}^{4}}{{({(x^{2} y)}^{\frac{1}{5}})}^{5}}$

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

$\frac{\sqrt{x \sqrt{y}} {\sqrt[5]{x^{2} y}}^{4}}{{(x^{2} y)}^{\frac{1}{5} \cdot 5}}$

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

$\frac{\sqrt{x \sqrt{y}} {\sqrt[5]{x^{2} y}}^{4}}{{(x^{2} y)}^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{\sqrt{x \sqrt{y}} {\sqrt[5]{x^{2} y}}^{4}}{{(x^{2} y)}^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{\sqrt{x \sqrt{y}} {\sqrt[5]{x^{2} y}}^{4}}{{(x^{2} y)}^{1}}$

Simplify.

$\frac{\sqrt{x \sqrt{y}} {\sqrt[5]{x^{2} y}}^{4}}{x^{2} y}$

Simplify the numerator.

Rewrite ${\sqrt[5]{x^{2} y}}^{4}$ as ${({(x^{2} y)}^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt{x \sqrt{y}} \sqrt[5]{{(x^{2} y)}^{4}}}{x^{2} y}$

Apply the product rule to $x^{2} y$ .

$\frac{\sqrt{x \sqrt{y}} \sqrt[5]{{(x^{2})}^{4} y^{4}}}{x^{2} y}$

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

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

$\frac{\sqrt{x \sqrt{y}} \sqrt[5]{x^{2 \cdot 4} y^{4}}}{x^{2} y}$

Multiply $2$ by $4$ .

$\frac{\sqrt{x \sqrt{y}} \sqrt[5]{x^{8} y^{4}}}{x^{2} y}$

Rewrite $x^{8} y^{4}$ as $x^{5} (x^{3} y^{4})$ .

Factor out $x^{5}$ .

$\frac{\sqrt{x \sqrt{y}} \sqrt[5]{x^{5} x^{3} y^{4}}}{x^{2} y}$

Add parentheses.

$\frac{\sqrt{x \sqrt{y}} \sqrt[5]{x^{5} (x^{3} y^{4})}}{x^{2} y}$

Pull terms out from under the radical.

$\frac{\sqrt{x \sqrt{y}} x \sqrt[5]{x^{3} y^{4}}}{x^{2} y}$

Combine exponents.

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

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

$\frac{x ({(x \sqrt{y})}^{\frac{1}{2}} \sqrt[5]{x^{3} y^{4}})}{x^{2} y}$

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

$\frac{x ({(x \sqrt{y})}^{\frac{5}{10}} \sqrt[5]{x^{3} y^{4}})}{x^{2} y}$

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

$\frac{x (\sqrt[10]{{(x \sqrt{y})}^{5}} \sqrt[5]{x^{3} y^{4}})}{x^{2} y}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{3} y^{4}}$ as ${(x^{3} y^{4})}^{\frac{1}{5}}$ .

$\frac{x (\sqrt[10]{{(x \sqrt{y})}^{5}} {(x^{3} y^{4})}^{\frac{1}{5}})}{x^{2} y}$

Rewrite ${(x^{3} y^{4})}^{\frac{1}{5}}$ as ${(x^{3} y^{4})}^{\frac{2}{10}}$ .

$\frac{x (\sqrt[10]{{(x \sqrt{y})}^{5}} {(x^{3} y^{4})}^{\frac{2}{10}})}{x^{2} y}$

Rewrite ${(x^{3} y^{4})}^{\frac{2}{10}}$ as $\sqrt[10]{{(x^{3} y^{4})}^{2}}$ .

$\frac{x (\sqrt[10]{{(x \sqrt{y})}^{5}} \sqrt[10]{{(x^{3} y^{4})}^{2}})}{x^{2} y}$

Combine using the product rule for radicals.

$\frac{x \sqrt[10]{{(x \sqrt{y})}^{5} {(x^{3} y^{4})}^{2}}}{x^{2} y}$

Apply the product rule to $x \sqrt{y}$ .

$\frac{x \sqrt[10]{x^{5} {\sqrt{y}}^{5} {(x^{3} y^{4})}^{2}}}{x^{2} y}$

Apply the product rule to $x^{3} y^{4}$ .

$\frac{x \sqrt[10]{x^{5} {\sqrt{y}}^{5} {(x^{3})}^{2} {(y^{4})}^{2}}}{x^{2} y}$

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

$\frac{x \sqrt[10]{x^{5} \sqrt{y^{5}} {(x^{3})}^{2} {(y^{4})}^{2}}}{x^{2} y}$

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

Factor out $y^{4}$ .

$\frac{x \sqrt[10]{x^{5} \sqrt{y^{4} y} {(x^{3})}^{2} {(y^{4})}^{2}}}{x^{2} y}$

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

$\frac{x \sqrt[10]{x^{5} \sqrt{{(y^{2})}^{2} y} {(x^{3})}^{2} {(y^{4})}^{2}}}{x^{2} y}$

Pull terms out from under the radical.

$\frac{x \sqrt[10]{x^{5} (y^{2} \sqrt{y}) {(x^{3})}^{2} {(y^{4})}^{2}}}{x^{2} y}$

Multiply the exponents in ${(x^{3})}^{2}$ .

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

$\frac{x \sqrt[10]{x^{5} (y^{2} \sqrt{y}) x^{3 \cdot 2} {(y^{4})}^{2}}}{x^{2} y}$

Multiply $3$ by $2$ .

$\frac{x \sqrt[10]{x^{5} (y^{2} \sqrt{y}) x^{6} {(y^{4})}^{2}}}{x^{2} y}$

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

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

$\frac{x \sqrt[10]{x^{5} (y^{2} \sqrt{y}) x^{6} y^{4 \cdot 2}}}{x^{2} y}$

Multiply $4$ by $2$ .

$\frac{x \sqrt[10]{x^{5} (y^{2} \sqrt{y}) x^{6} y^{8}}}{x^{2} y}$

$\frac{x \sqrt[10]{x^{5} y^{2} \sqrt{y} x^{6} y^{8}}}{x^{2} y}$

Combine exponents.

Multiply $x^{5}$ by $x^{6}$ by adding the exponents.

Move $x^{6}$ .

$\frac{x \sqrt[10]{x^{6} x^{5} y^{2} \sqrt{y} y^{8}}}{x^{2} y}$

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

$\frac{x \sqrt[10]{x^{6 + 5} y^{2} \sqrt{y} y^{8}}}{x^{2} y}$

Add $6$ and $5$ .

$\frac{x \sqrt[10]{x^{11} y^{2} \sqrt{y} y^{8}}}{x^{2} y}$

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

Move $y^{8}$ .

$\frac{x \sqrt[10]{x^{11} (y^{8} y^{2}) \sqrt{y}}}{x^{2} y}$

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

$\frac{x \sqrt[10]{x^{11} y^{8 + 2} \sqrt{y}}}{x^{2} y}$

Add $8$ and $2$ .

$\frac{x \sqrt[10]{x^{11} y^{10} \sqrt{y}}}{x^{2} y}$

Rewrite $x^{11} y^{10} \sqrt{y}$ as ${(x y)}^{10} (x \sqrt{y})$ .

Factor out $x^{10}$ .

$\frac{x \sqrt[10]{x^{10} x y^{10} \sqrt{y}}}{x^{2} y}$

Move $x$ .

$\frac{x \sqrt[10]{x^{10} y^{10} x \sqrt{y}}}{x^{2} y}$

Rewrite $x^{10} y^{10}$ as ${(x y)}^{10}$ .

$\frac{x \sqrt[10]{{(x y)}^{10} x \sqrt{y}}}{x^{2} y}$

Add parentheses.

$\frac{x \sqrt[10]{{(x y)}^{10} (x \sqrt{y})}}{x^{2} y}$

Pull terms out from under the radical.

$\frac{x (x y \sqrt[10]{x \sqrt{y}})}{x^{2} y}$

Simplify the numerator.

Raise $x$ to the power of $1$ .

$\frac{x^{1} x (y \sqrt[10]{x \sqrt{y}})}{x^{2} y}$

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{1} (y \sqrt[10]{x \sqrt{y}})}{x^{2} y}$

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

$\frac{x^{1 + 1} (y \sqrt[10]{x \sqrt{y}})}{x^{2} y}$

Add $1$ and $1$ .

$\frac{x^{2} (y \sqrt[10]{x \sqrt{y}})}{x^{2} y}$

Simplify the numerator.

Rewrite.