Simplify the Radical Expression ( cube root of x square root of y)/( fourth root of xy^2)

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

Step 1

Multiply $\frac{\sqrt[3]{x \sqrt{y}}}{\sqrt[4]{x y^{2}}}$ by $\frac{{\sqrt[4]{x y^{2}}}^{3}}{{\sqrt[4]{x y^{2}}}^{3}}$ .

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

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{x \sqrt{y}}}{\sqrt[4]{x y^{2}}}$ by $\frac{{\sqrt[4]{x y^{2}}}^{3}}{{\sqrt[4]{x y^{2}}}^{3}}$ .

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

Raise $\sqrt[4]{x y^{2}}$ to the power of $1$ .

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

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

$\frac{\sqrt[3]{x \sqrt{y}} {\sqrt[4]{x y^{2}}}^{3}}{{\sqrt[4]{x y^{2}}}^{1 + 3}}$

Add $1$ and $3$ .

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

Rewrite ${\sqrt[4]{x y^{2}}}^{4}$ as $x y^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x y^{2}}$ as ${(x y^{2})}^{\frac{1}{4}}$ .

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

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

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

Combine $\frac{1}{4}$ and $4$ .

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 3

Simplify the numerator.

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

$\frac{\sqrt[3]{x \sqrt{y}} \sqrt[4]{x^{3} y^{6}}}{x y^{2}}$

Rewrite $x^{3} y^{6}$ as $y^{4} (x^{3} y^{2})$ .

Factor out $y^{4}$ .

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

Reorder $x^{3}$ and $y^{4}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

Combine exponents.

Rewrite the expression using the least common index of $12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3} y^{2}}$ as ${(x^{3} y^{2})}^{\frac{1}{4}}$ .

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

Rewrite ${(x^{3} y^{2})}^{\frac{1}{4}}$ as ${(x^{3} y^{2})}^{\frac{3}{12}}$ .

$\frac{{(x^{3} y^{2})}^{\frac{3}{12}} \sqrt[3]{x \sqrt{y}} | y |}{x y^{2}}$

Rewrite ${(x^{3} y^{2})}^{\frac{3}{12}}$ as $\sqrt[12]{{(x^{3} y^{2})}^{3}}$ .

$\frac{\sqrt[12]{{(x^{3} y^{2})}^{3}} \sqrt[3]{x \sqrt{y}} | y |}{x y^{2}}$

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

$\frac{\sqrt[12]{{(x^{3} y^{2})}^{3}} {(x \sqrt{y})}^{\frac{1}{3}} | y |}{x y^{2}}$

Rewrite ${(x \sqrt{y})}^{\frac{1}{3}}$ as ${(x \sqrt{y})}^{\frac{4}{12}}$ .

$\frac{\sqrt[12]{{(x^{3} y^{2})}^{3}} {(x \sqrt{y})}^{\frac{4}{12}} | y |}{x y^{2}}$

Rewrite ${(x \sqrt{y})}^{\frac{4}{12}}$ as $\sqrt[12]{{(x \sqrt{y})}^{4}}$ .

$\frac{\sqrt[12]{{(x^{3} y^{2})}^{3}} \sqrt[12]{{(x \sqrt{y})}^{4}} | y |}{x y^{2}}$

Combine using the product rule for radicals.

$\frac{\sqrt[12]{{(x^{3} y^{2})}^{3} {(x \sqrt{y})}^{4}} | y |}{x y^{2}}$

Apply the product rule to $x^{3} y^{2}$ .

$\frac{\sqrt[12]{{(x^{3})}^{3} {(y^{2})}^{3} {(x \sqrt{y})}^{4}} | y |}{x y^{2}}$

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

$\frac{\sqrt[12]{{(x^{3})}^{3} {(y^{2})}^{3} x^{4} {\sqrt{y}}^{4}} | y |}{x y^{2}}$

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

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

$\frac{\sqrt[12]{x^{3 \cdot 3} {(y^{2})}^{3} x^{4} {\sqrt{y}}^{4}} | y |}{x y^{2}}$

Multiply $3$ by $3$ .

$\frac{\sqrt[12]{x^{9} {(y^{2})}^{3} x^{4} {\sqrt{y}}^{4}} | y |}{x y^{2}}$

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

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

$\frac{\sqrt[12]{x^{9} y^{2 \cdot 3} x^{4} {\sqrt{y}}^{4}} | y |}{x y^{2}}$

Multiply $2$ by $3$ .

$\frac{\sqrt[12]{x^{9} y^{6} x^{4} {\sqrt{y}}^{4}} | y |}{x y^{2}}$

Rewrite ${\sqrt{y}}^{4}$ as $y^{2}$ .

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

$\frac{\sqrt[12]{x^{9} y^{6} x^{4} {(y^{\frac{1}{2}})}^{4}} | y |}{x y^{2}}$

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

$\frac{\sqrt[12]{x^{9} y^{6} x^{4} y^{\frac{1}{2} \cdot 4}} | y |}{x y^{2}}$

Combine $\frac{1}{2}$ and $4$ .

$\frac{\sqrt[12]{x^{9} y^{6} x^{4} y^{\frac{4}{2}}} | y |}{x y^{2}}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$\frac{\sqrt[12]{x^{9} y^{6} x^{4} y^{\frac{2 \cdot 2}{2}}} | y |}{x y^{2}}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{\sqrt[12]{x^{9} y^{6} x^{4} y^{\frac{2 \cdot 2}{2 (1)}}} | y |}{x y^{2}}$

Cancel the common factor.

$\frac{\sqrt[12]{x^{9} y^{6} x^{4} y^{\frac{2 \cdot 2}{2 \cdot 1}}} | y |}{x y^{2}}$

Rewrite the expression.

$\frac{\sqrt[12]{x^{9} y^{6} x^{4} y^{\frac{2}{1}}} | y |}{x y^{2}}$

Divide $2$ by $1$ .

$\frac{\sqrt[12]{x^{9} y^{6} x^{4} y^{2}} | y |}{x y^{2}}$

Combine exponents.

Multiply $x^{9}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$\frac{\sqrt[12]{x^{4} x^{9} y^{6} y^{2}} | y |}{x y^{2}}$

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

$\frac{\sqrt[12]{x^{4 + 9} y^{6} y^{2}} | y |}{x y^{2}}$

Add $4$ and $9$ .

$\frac{\sqrt[12]{x^{13} y^{6} y^{2}} | y |}{x y^{2}}$

Multiply $y^{6}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$\frac{\sqrt[12]{x^{13} (y^{2} y^{6})} | y |}{x y^{2}}$

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

$\frac{\sqrt[12]{x^{13} y^{2 + 6}} | y |}{x y^{2}}$

Add $2$ and $6$ .

$\frac{\sqrt[12]{x^{13} y^{8}} | y |}{x y^{2}}$

Rewrite $x^{13} y^{8}$ as $x^{12} (x y^{8})$ .

Factor out $x^{12}$ .

$\frac{\sqrt[12]{x^{12} x y^{8}} | y |}{x y^{2}}$

Add parentheses.

$\frac{\sqrt[12]{x^{12} (x y^{8})} | y |}{x y^{2}}$

Pull terms out from under the radical.

$\frac{| x | \sqrt[12]{x y^{8}} | y |}{x y^{2}}$

Step 4

To multiply absolute values, multiply the terms inside each absolute value.

$\frac{| y x | \sqrt[12]{x y^{8}}}{x y^{2}}$

Simplify the Radical Expression ( cube root of x square root of y)/( fourth root of xy^2)

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