Algebra Examples

$(\sqrt{x} - \sqrt{2 y}) (\sqrt{3 y} + \sqrt{2 x})$

Step 1

Expand $(\sqrt{x} - \sqrt{2 y}) (\sqrt{3 y} + \sqrt{2 x})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{x} (\sqrt{3 y} + \sqrt{2 x}) - \sqrt{2 y} (\sqrt{3 y} + \sqrt{2 x})$

Step 1.2

Apply the distributive property.

$\sqrt{x} \sqrt{3 y} + \sqrt{x} \sqrt{2 x} - \sqrt{2 y} (\sqrt{3 y} + \sqrt{2 x})$

Step 1.3

Apply the distributive property.

$\sqrt{x} \sqrt{3 y} + \sqrt{x} \sqrt{2 x} - \sqrt{2 y} \sqrt{3 y} - \sqrt{2 y} \sqrt{2 x}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine using the product rule for radicals.

$\sqrt{x \cdot 3 y} + \sqrt{x} \sqrt{2 x} - \sqrt{2 y} \sqrt{3 y} - \sqrt{2 y} \sqrt{2 x}$

Step 2.1.2

Multiply $\sqrt{x} \sqrt{2 x}$ .

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Step 2.1.2.1

Combine using the product rule for radicals.

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

Add $1$ and $1$ .

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

Step 2.1.3

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

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

Step 2.1.4

Pull terms out from under the radical.

$\sqrt{x \cdot 3 y} + x \sqrt{2} - \sqrt{2 y} \sqrt{3 y} - \sqrt{2 y} \sqrt{2 x}$

Step 2.1.5

Multiply $- \sqrt{2 y} \sqrt{3 y}$ .

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Step 2.1.5.1

Combine using the product rule for radicals.

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

Step 2.1.5.2

Multiply $2$ by $3$ .

$\sqrt{x \cdot 3 y} + x \sqrt{2} - \sqrt{6 y \cdot y} - \sqrt{2 y} \sqrt{2 x}$

Step 2.1.5.3

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

$\sqrt{x \cdot 3 y} + x \sqrt{2} - \sqrt{6 (y^{1} y)} - \sqrt{2 y} \sqrt{2 x}$

Step 2.1.5.4

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

$\sqrt{x \cdot 3 y} + x \sqrt{2} - \sqrt{6 (y^{1} y^{1})} - \sqrt{2 y} \sqrt{2 x}$

Step 2.1.5.5

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

$\sqrt{x \cdot 3 y} + x \sqrt{2} - \sqrt{6 y^{1 + 1}} - \sqrt{2 y} \sqrt{2 x}$

Step 2.1.5.6

Add $1$ and $1$ .

$\sqrt{x \cdot 3 y} + x \sqrt{2} - \sqrt{6 y^{2}} - \sqrt{2 y} \sqrt{2 x}$

Step 2.1.6

Reorder $6$ and $y^{2}$ .

$\sqrt{x \cdot 3 y} + x \sqrt{2} - \sqrt{y^{2} \cdot 6} - \sqrt{2 y} \sqrt{2 x}$

Step 2.1.7

Pull terms out from under the radical.

$\sqrt{x \cdot 3 y} + x \sqrt{2} - (y \sqrt{6}) - \sqrt{2 y} \sqrt{2 x}$

Step 2.1.8

Multiply $- \sqrt{2 y} \sqrt{2 x}$ .

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Step 2.1.8.1

Combine using the product rule for radicals.

$\sqrt{x \cdot 3 y} + x \sqrt{2} - y \sqrt{6} - \sqrt{2 x (2 y)}$

Step 2.1.8.2

Multiply $2$ by $2$ .

$\sqrt{x \cdot 3 y} + x \sqrt{2} - y \sqrt{6} - \sqrt{4 x y}$

Step 2.1.9

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

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Step 2.1.9.1

Rewrite $4$ as $2^{2}$ .

$\sqrt{x \cdot 3 y} + x \sqrt{2} - y \sqrt{6} - \sqrt{2^{2} x y}$

Step 2.1.9.2

Add parentheses.

$\sqrt{x \cdot 3 y} + x \sqrt{2} - y \sqrt{6} - \sqrt{2^{2} (x y)}$

Step 2.1.10

Pull terms out from under the radical.

$\sqrt{x \cdot 3 y} + x \sqrt{2} - y \sqrt{6} - (2 \sqrt{x y})$

Step 2.1.11

Multiply $2$ by $- 1$ .

$\sqrt{x \cdot 3 y} + x \sqrt{2} - y \sqrt{6} - 2 \sqrt{x y}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Reorder factors in $\sqrt{x \cdot 3 y} + x \sqrt{2} - y \sqrt{6} - 2 \sqrt{x y}$ .

$\sqrt{3 x y} + x \sqrt{2} - y \sqrt{6} - 2 \sqrt{x y}$

Step 2.2.2

Move $\sqrt{3 x y}$ .

$x \sqrt{2} - y \sqrt{6} + \sqrt{3 x y} - 2 \sqrt{x y}$