Basic Math Examples

$\frac{1 + \sqrt{6}}{\sqrt{3} + \sqrt{2}}$

Step 1

Multiply $\frac{1 + \sqrt{6}}{\sqrt{3} + \sqrt{2}}$ by $\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{1 + \sqrt{6}}{\sqrt{3} + \sqrt{2}}$ by $\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 2.4

Divide $(1 + \sqrt{6}) (\sqrt{3} - \sqrt{2})$ by $1$ .

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

Step 3

Expand $(1 + \sqrt{6}) (\sqrt{3} - \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

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

Multiply $\sqrt{3}$ by $1$ .

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

Step 4.1.2

Multiply $- \sqrt{2}$ by $1$ .

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

Step 4.1.3

Combine using the product rule for radicals.

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

Step 4.1.4

Multiply $6$ by $3$ .

$\sqrt{3} - \sqrt{2} + \sqrt{18} + \sqrt{6} (- \sqrt{2})$

Step 4.1.5

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.1.5.1

Factor $9$ out of $18$ .

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

Step 4.1.5.2

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

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

Step 4.1.6

Pull terms out from under the radical.

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

Step 4.1.7

Multiply $\sqrt{6} (- \sqrt{2})$ .

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Step 4.1.7.1

Combine using the product rule for radicals.

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

Step 4.1.7.2

Multiply $6$ by $2$ .

$\sqrt{3} - \sqrt{2} + 3 \sqrt{2} - \sqrt{12}$

Step 4.1.8

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.1.8.1

Factor $4$ out of $12$ .

$\sqrt{3} - \sqrt{2} + 3 \sqrt{2} - \sqrt{4 (3)}$

Step 4.1.8.2

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

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

Step 4.1.9

Pull terms out from under the radical.

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

Step 4.1.10

Multiply $2$ by $- 1$ .

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

Step 4.2

Subtract $2 \sqrt{3}$ from $\sqrt{3}$ .

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

Step 4.3

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.09637631 \dots$