Algebra Examples

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

Step 1

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

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Step 1.1

Factor out $y^{2}$ .

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

Step 1.2

Reorder $x$ and $y^{2}$ .

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

Step 1.3

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

Rewrite the expression using the least common index of $6$ .

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Step 3.1.1

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

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

Step 3.1.2

Rewrite ${(x^{2} y)}^{\frac{1}{3}}$ as ${(x^{2} y)}^{\frac{2}{6}}$ .

$y ({(x^{2} y)}^{\frac{2}{6}} \sqrt{x y})$

Step 3.1.3

Rewrite ${(x^{2} y)}^{\frac{2}{6}}$ as $\sqrt[6]{{(x^{2} y)}^{2}}$ .

$y (\sqrt[6]{{(x^{2} y)}^{2}} \sqrt{x y})$

Step 3.1.4

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

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

Step 3.1.5

Rewrite ${(x y)}^{\frac{1}{2}}$ as ${(x y)}^{\frac{3}{6}}$ .

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

Step 3.1.6

Rewrite ${(x y)}^{\frac{3}{6}}$ as $\sqrt[6]{{(x y)}^{3}}$ .

$y (\sqrt[6]{{(x^{2} y)}^{2}} \sqrt[6]{{(x y)}^{3}})$

Step 3.2

Combine using the product rule for radicals.

$y \sqrt[6]{{(x^{2} y)}^{2} {(x y)}^{3}}$

Step 3.3

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

$y \sqrt[6]{{(x^{2})}^{2} y^{2} {(x y)}^{3}}$

Step 3.4

Apply the product rule to $x y$ .

$y \sqrt[6]{{(x^{2})}^{2} y^{2} x^{3} y^{3}}$

Step 3.5

Multiply $y^{2}$ by $y^{3}$ by adding the exponents.

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Step 3.5.1

Move $y^{3}$ .

$y \sqrt[6]{{(x^{2})}^{2} (y^{3} y^{2}) x^{3}}$

Step 3.5.2

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

$y \sqrt[6]{{(x^{2})}^{2} y^{3 + 2} x^{3}}$

Step 3.5.3

Add $3$ and $2$ .

$y \sqrt[6]{{(x^{2})}^{2} y^{5} x^{3}}$

Step 3.6

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

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Step 3.6.1

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

$y \sqrt[6]{x^{2 \cdot 2} y^{5} x^{3}}$

Step 3.6.2

Multiply $2$ by $2$ .

$y \sqrt[6]{x^{4} y^{5} x^{3}}$

Step 3.7

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 3.7.1

Move $x^{3}$ .

$y \sqrt[6]{x^{3} x^{4} y^{5}}$

Step 3.7.2

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

$y \sqrt[6]{x^{3 + 4} y^{5}}$

Step 3.7.3

Add $3$ and $4$ .

$y \sqrt[6]{x^{7} y^{5}}$

Step 4

Rewrite $x^{7} y^{5}$ as $x^{6} (x y^{5})$ .

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Step 4.1

Factor out $x^{6}$ .

$y \sqrt[6]{x^{6} x y^{5}}$

Step 4.2

Add parentheses.

$y \sqrt[6]{x^{6} (x y^{5})}$

Step 5

Pull terms out from under the radical.

$y (x \sqrt[6]{x y^{5}})$

Step 6

Reorder factors in $y (x \sqrt[6]{x y^{5}})$ .

$y x \sqrt[6]{x y^{5}}$