Algebra Examples

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

Step 1

Rewrite $2 x^{6} y^{5}$ as ${(x^{3} y^{2})}^{2} \cdot (2 y)$ .

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Step 1.1

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

$\sqrt{2 {(x^{3})}^{2} y^{5}} \cdot \sqrt{2 x^{2} y}$

Step 1.2

Factor out $y^{4}$ .

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

Step 1.3

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

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

Step 1.4

Move $2$ .

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

Step 1.5

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

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

Step 1.6

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Move $x$ .

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

Step 5.2

Multiply $x$ by $x^{3}$ .

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Step 5.2.1

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

$x^{1} x^{3} y^{2} \sqrt{2 y} \cdot \sqrt{2 y}$

Step 5.2.2

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

$x^{1 + 3} y^{2} \sqrt{2 y} \cdot \sqrt{2 y}$

Step 5.3

Add $1$ and $3$ .

$x^{4} y^{2} \sqrt{2 y} \cdot \sqrt{2 y}$

Step 6

Multiply $x^{4} y^{2} \sqrt{2 y} \sqrt{2 y}$ .

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Step 6.1

Raise $\sqrt{2 y}$ to the power of $1$ .

$x^{4} y^{2} ({\sqrt{2 y}}^{1} \sqrt{2 y})$

Step 6.2

Raise $\sqrt{2 y}$ to the power of $1$ .

$x^{4} y^{2} ({\sqrt{2 y}}^{1} {\sqrt{2 y}}^{1})$

Step 6.3

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

$x^{4} y^{2} {\sqrt{2 y}}^{1 + 1}$

Step 6.4

Add $1$ and $1$ .

$x^{4} y^{2} {\sqrt{2 y}}^{2}$

Step 7

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

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Step 7.1

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

$x^{4} y^{2} {({(2 y)}^{\frac{1}{2}})}^{2}$

Step 7.2

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

$x^{4} y^{2} {(2 y)}^{\frac{1}{2} \cdot 2}$

Step 7.3

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

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

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

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

Step 7.4.2

Rewrite the expression.

$x^{4} y^{2} {(2 y)}^{1}$

Step 7.5

Simplify.

$x^{4} y^{2} (2 y)$

Step 8

Rewrite using the commutative property of multiplication.

$2 x^{4} y^{2} y$

Step 9

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

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Step 9.1

Move $y$ .

$2 x^{4} (y \cdot y^{2})$

Step 9.2

Multiply $y$ by $y^{2}$ .

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Step 9.2.1

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

$2 x^{4} (y^{1} y^{2})$

Step 9.2.2

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

$2 x^{4} y^{1 + 2}$

Step 9.3

Add $1$ and $2$ .

$2 x^{4} y^{3}$