Basic Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $2$ by $3$ .

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

Step 6.1.2

Move $3$ to the left of $\sqrt{3}$ .

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

Step 6.1.3

Combine using the product rule for radicals.

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

Step 6.1.4

Multiply $3$ by $3$ .

$\frac{6 + 2 \sqrt{3} + 3 \sqrt{3} + \sqrt{9}}{6}$

Step 6.1.5

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

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

Step 6.1.6

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

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

Step 6.2

Add $6$ and $3$ .

$\frac{9 + 2 \sqrt{3} + 3 \sqrt{3}}{6}$

Step 6.3

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

$\frac{9 + 5 \sqrt{3}}{6}$

Step 7

Split the fraction $\frac{9 + 5 \sqrt{3}}{6}$ into two fractions.

$\frac{9}{6} + \frac{5 \sqrt{3}}{6}$

Step 8

Cancel the common factor of $9$ and $6$ .

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Step 8.1

Factor $3$ out of $9$ .

$\frac{3 (3)}{6} + \frac{5 \sqrt{3}}{6}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 3}{3 \cdot 2} + \frac{5 \sqrt{3}}{6}$

Step 8.2.2

Cancel the common factor.

$\frac{3 \cdot 3}{3 \cdot 2} + \frac{5 \sqrt{3}}{6}$

Step 8.2.3

Rewrite the expression.

$\frac{3}{2} + \frac{5 \sqrt{3}}{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2} + \frac{5 \sqrt{3}}{6}$

Decimal Form:

$2.94337567 \dots$