Pre-Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\sqrt{3 x} \sqrt{3 x} + \sqrt{3 x} (- \sqrt{2 y}) + \sqrt{2 y} \sqrt{3 x} + \sqrt{2 y} (- \sqrt{2 y})$ .

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Step 2.1.1

Reorder the factors in the terms $\sqrt{3 x} (- \sqrt{2 y})$ and $\sqrt{2 y} \sqrt{3 x}$ .

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

Step 2.1.2

Add $- \sqrt{2 y} \sqrt{3 x}$ and $\sqrt{2 y} \sqrt{3 x}$ .

$\sqrt{3 x} \sqrt{3 x} + 0 + \sqrt{2 y} (- \sqrt{2 y})$

Step 2.1.3

Add $\sqrt{3 x} \sqrt{3 x}$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $\sqrt{3 x} \sqrt{3 x}$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Add $1$ and $1$ .

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

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

Step 2.2.2.4.2

Rewrite the expression.

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

Step 2.2.2.5

Simplify.

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

Step 2.2.3

Rewrite using the commutative property of multiplication.

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

Step 2.2.4

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

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Step 2.2.4.1

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

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

Step 2.2.4.2

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

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

Step 2.2.4.3

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

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

Step 2.2.4.4

Add $1$ and $1$ .

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

Step 2.2.5

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

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Step 2.2.5.1

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

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

Step 2.2.5.2

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

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

Step 2.2.5.3

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

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

Step 2.2.5.4

Cancel the common factor of $2$ .

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Step 2.2.5.4.1

Cancel the common factor.

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

Step 2.2.5.4.2

Rewrite the expression.

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

Step 2.2.5.5

Simplify.

$3 x - (2 y)$

Step 2.2.6

Multiply $2$ by $- 1$ .

$3 x - 2 y$