Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like 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.

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

Step 2.1.2

Multiply $3$ by $3$ .

$\sqrt{9} + \sqrt{3} (- \sqrt{2}) + \sqrt{2} \sqrt{3} + \sqrt{2} (- \sqrt{2})$

Step 2.1.3

Rewrite $9$ as $3^{2}$ .

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

Step 2.1.4

Pull terms out from under the radical, assuming positive real numbers.

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

Step 2.1.5

Multiply $\sqrt{3} (- \sqrt{2})$ .

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Step 2.1.5.1

Combine using the product rule for radicals.

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

Step 2.1.5.2

Multiply $3$ by $2$ .

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

Step 2.1.6

Combine using the product rule for radicals.

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

Step 2.1.7

Multiply $2$ by $3$ .

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

Step 2.1.8

Multiply $\sqrt{2} (- \sqrt{2})$ .

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Step 2.1.8.1

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

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

Step 2.1.8.2

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

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

Step 2.1.8.3

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

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

Step 2.1.8.4

Add $1$ and $1$ .

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

Step 2.1.9

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

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Step 2.1.9.1

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

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

Step 2.1.9.2

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

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

Step 2.1.9.3

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

$3 - \sqrt{6} + \sqrt{6} - 2^{\frac{2}{2}}$

Step 2.1.9.4

Cancel the common factor of $2$ .

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Step 2.1.9.4.1

Cancel the common factor.

$3 - \sqrt{6} + \sqrt{6} - 2^{\frac{2}{2}}$

Step 2.1.9.4.2

Rewrite the expression.

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

Step 2.1.9.5

Evaluate the exponent.

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

Step 2.1.10

Multiply $- 1$ by $2$ .

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

Step 2.2

Subtract $2$ from $3$ .

$1 - \sqrt{6} + \sqrt{6}$

Step 2.3

Add $- \sqrt{6}$ and $\sqrt{6}$ .

$1 + 0$

Step 2.4

Add $1$ and $0$ .

$1$