Algebra Examples

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

Step 1

Multiply $\frac{2 + \sqrt{3}}{5 - \sqrt{3}}$ by $\frac{5 + \sqrt{3}}{5 + \sqrt{3}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{2 + \sqrt{3}}{5 - \sqrt{3}}$ by $\frac{5 + \sqrt{3}}{5 + \sqrt{3}}$ .

$\frac{(2 + \sqrt{3}) (5 + \sqrt{3})}{(5 - \sqrt{3}) (5 + \sqrt{3})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(2 + \sqrt{3}) (5 + \sqrt{3})}{25 + 5 \sqrt{3} - 5 \sqrt{3} - {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

$\frac{(2 + \sqrt{3}) (5 + \sqrt{3})}{22}$

Step 3

Expand $(2 + \sqrt{3}) (5 + \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{2 (5 + \sqrt{3}) + \sqrt{3} (5 + \sqrt{3})}{22}$

Step 3.2

Apply the distributive property.

$\frac{2 \cdot 5 + 2 \sqrt{3} + \sqrt{3} (5 + \sqrt{3})}{22}$

Step 3.3

Apply the distributive property.

$\frac{2 \cdot 5 + 2 \sqrt{3} + \sqrt{3} \cdot 5 + \sqrt{3} \sqrt{3}}{22}$

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 $2$ by $5$ .

$\frac{10 + 2 \sqrt{3} + \sqrt{3} \cdot 5 + \sqrt{3} \sqrt{3}}{22}$

Step 4.1.2

Move $5$ to the left of $\sqrt{3}$ .

$\frac{10 + 2 \sqrt{3} + 5 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{22}$

Step 4.1.3

Combine using the product rule for radicals.

$\frac{10 + 2 \sqrt{3} + 5 \sqrt{3} + \sqrt{3 \cdot 3}}{22}$

Step 4.1.4

Multiply $3$ by $3$ .

$\frac{10 + 2 \sqrt{3} + 5 \sqrt{3} + \sqrt{9}}{22}$

Step 4.1.5

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

$\frac{10 + 2 \sqrt{3} + 5 \sqrt{3} + \sqrt{3^{2}}}{22}$

Step 4.1.6

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

$\frac{10 + 2 \sqrt{3} + 5 \sqrt{3} + 3}{22}$

Step 4.2

Add $10$ and $3$ .

$\frac{13 + 2 \sqrt{3} + 5 \sqrt{3}}{22}$

Step 4.3

Add $2 \sqrt{3}$ and $5 \sqrt{3}$ .

$\frac{13 + 7 \sqrt{3}}{22}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{13 + 7 \sqrt{3}}{22}$

Decimal Form:

$1.14201616 \dots$