Algebra Examples

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

Step 1

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

Tap for more steps...

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

Tap for more steps...

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

Tap for more steps...

Step 6.1

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

Tap for more steps...

Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

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

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

Step 6.1.5

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

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

Step 6.1.6

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

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

Step 6.2

Combine using the product rule for radicals.

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

Step 6.3

Apply the product rule to $3 y$ .

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

Step 6.4

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

Tap for more steps...

Step 6.4.1

Move $y^{4}$ .

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

Step 6.4.2

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

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

Step 6.4.3

Add $4$ and $3$ .

$3 | x | (x \sqrt[6]{3^{3} y^{7}})$

Step 6.5

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

$3 | x | (x \sqrt[6]{27 y^{7}})$

Step 7

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

Tap for more steps...

Step 7.1

Factor out $y^{6}$ .

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

Step 7.2

Reorder $27$ and $y^{6}$ .

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

Step 7.3

Add parentheses.

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

Step 8

Pull terms out from under the radical.

$3 | x | (x (| y | \sqrt[6]{27 y}))$

Step 9

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

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