Algebra Examples

$\frac{2 + \sqrt{3}}{\sqrt{3} + \sqrt{2}}$

Step 1

Multiply $\frac{2 + \sqrt{3}}{\sqrt{3} + \sqrt{2}}$ by $\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{2 + \sqrt{3}}{\sqrt{3} + \sqrt{2}}$ by $\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 2.4

Divide $(2 + \sqrt{3}) (\sqrt{3} - \sqrt{2})$ by $1$ .

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify each term.

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Step 4.1

Multiply $- 1$ by $2$ .

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

Step 4.2

Combine using the product rule for radicals.

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

Step 4.3

Multiply $3$ by $3$ .

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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

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Step 4.6.1

Combine using the product rule for radicals.

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

Step 4.6.2

Multiply $3$ by $2$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.18618474 \dots$