Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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.

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

Step 2.1.2

Multiply $2$ by $3$ .

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

Add $1$ and $1$ .

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

Step 2.1.4

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

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Step 2.1.4.1

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

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

Step 2.1.4.2

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

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

Step 2.1.4.3

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

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

Step 2.1.4.4

Cancel the common factor of $2$ .

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Step 2.1.4.4.1

Cancel the common factor.

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

Step 2.1.4.4.2

Rewrite the expression.

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

Step 2.1.4.5

Evaluate the exponent.

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

Step 2.1.5

Multiply $- 1$ by $2$ .

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

Step 2.1.6

Combine using the product rule for radicals.

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

Step 2.1.7

Multiply $3$ by $3$ .

$\sqrt{6} - 2 + \sqrt{9} + \sqrt{3} (- \sqrt{2})$

Step 2.1.8

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

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

Step 2.1.9

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

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

Step 2.1.10

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

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Step 2.1.10.1

Combine using the product rule for radicals.

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

Step 2.1.10.2

Multiply $3$ by $2$ .

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

Step 2.2

Subtract $\sqrt{6}$ from $\sqrt{6}$ .

$0 - 2 + 3$

Step 2.3

Subtract $2$ from $0$ .

$- 2 + 3$

Step 2.4

Add $- 2$ and $3$ .

$1$