Simplify ( square root of x- square root of 2y)( square root of 3y+ square root of 2x)

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

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.

Simplify each term.

Combine using the product rule for radicals.

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

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

Combine using the product rule for radicals.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Pull terms out from under the radical.

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Pull terms out from under the radical.

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

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

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

Simplify the expression.

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

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

Move $\sqrt{3 x y}$ .

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

Simplify ( square root of x- square root of 2y)( square root of 3y+ square root of 2x)

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