Algebra Examples

$\sqrt{3 x^{2} y} \cdot \sqrt[3]{27 x^{3} y^{2}}$

Step 1

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

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Step 1.1

Reorder $3$ and $x^{2}$ .

$\sqrt{x^{2} \cdot 3 y} \cdot \sqrt[3]{27 x^{3} y^{2}}$

Step 1.2

Add parentheses.

$\sqrt{x^{2} \cdot (3 y)} \cdot \sqrt[3]{27 x^{3} y^{2}}$

Step 2

Pull terms out from under the radical.

$x \sqrt{3 y} \cdot \sqrt[3]{27 x^{3} y^{2}}$

Step 3

Rewrite $27 x^{3} y^{2}$ as ${(3 x)}^{3} y^{2}$ .

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Step 3.1

Rewrite $27$ as $3^{3}$ .

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

Step 3.2

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

Multiply $x$ by $x$ by adding the exponents.

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Step 6.1

Move $x$ .

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 7

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

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

Rewrite $y^{\frac{2}{3}}$ as $y^{\frac{4}{6}}$ .

$3 x^{2} (y^{\frac{4}{6}} \sqrt{3 y})$

Step 7.1.3

Rewrite $y^{\frac{4}{6}}$ as $\sqrt[6]{y^{4}}$ .

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

Step 7.1.4

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

$3 x^{2} (\sqrt[6]{y^{4}} {(3 y)}^{\frac{1}{2}})$

Step 7.1.5

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

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

Step 7.1.6

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

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

Step 7.2

Combine using the product rule for radicals.

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

Step 7.3

Apply the product rule to $3 y$ .

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

Step 7.4

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

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Step 7.4.1

Move $y^{3}$ .

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

Step 7.4.2

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

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

Step 7.4.3

Add $3$ and $4$ .

$3 x^{2} \sqrt[6]{y^{7} \cdot 3^{3}}$

Step 7.5

Raise $3$ to the power of $3$ .

$3 x^{2} \sqrt[6]{y^{7} \cdot 27}$

Step 8

Rewrite $y^{7} \cdot 27$ as $y^{6} (y \cdot 27)$ .

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Step 8.1

Factor out $y^{6}$ .

$3 x^{2} \sqrt[6]{y^{6} y \cdot 27}$

Step 8.2

Add parentheses.

$3 x^{2} \sqrt[6]{y^{6} (y \cdot 27)}$

Step 9

Pull terms out from under the radical.

$3 x^{2} (y \sqrt[6]{y \cdot 27})$

Step 10

Reorder factors in $3 x^{2} y \sqrt[6]{y \cdot 27}$ .

$3 x^{2} y \sqrt[6]{27 y}$