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

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

Step 1

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

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Rewrite using the commutative property of multiplication.

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

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

Multiply $- \sqrt{3 y} (- \sqrt{3 y})$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\sqrt{3 y}$ by $1$ .

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

Raise $\sqrt{3 y}$ to the power of $1$ .

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

Raise $\sqrt{3 y}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3 y}}^{2}$ as $3 y$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Subtract $\sqrt{6 x y}$ from $- \sqrt{6 y x}$ .

Move $y$ .

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

Subtract $\sqrt{6 x y}$ from $- \sqrt{6 x y}$ .

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

Step 4

Move $- 2 \sqrt{6 x y}$ .

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

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

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