Basic Math Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 2.4

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

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Combine using the product rule for radicals.

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

Step 4.1.2

Multiply $3$ by $3$ .

$\sqrt{9} + \sqrt{3} \sqrt{2} + \sqrt{2} \sqrt{3} + \sqrt{2} \sqrt{2}$

Step 4.1.3

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

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

Step 4.1.4

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 4.1.5

Combine using the product rule for radicals.

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

Step 4.1.6

Multiply $3$ by $2$ .

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

Step 4.1.7

Combine using the product rule for radicals.

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

Step 4.1.8

Multiply $2$ by $3$ .

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

Step 4.1.9

Combine using the product rule for radicals.

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

Step 4.1.10

Multiply $2$ by $2$ .

$3 + \sqrt{6} + \sqrt{6} + \sqrt{4}$

Step 4.1.11

Rewrite $4$ as $2^{2}$ .

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

Step 4.1.12

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

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

Step 4.2

Add $3$ and $2$ .

$5 + \sqrt{6} + \sqrt{6}$

Step 4.3

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

$5 + 2 \sqrt{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$5 + 2 \sqrt{6}$

Decimal Form:

$9.89897948 \dots$