Algebra Examples

$\frac{4 + \sqrt{12}}{4 - \sqrt{12}}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $12$ .

$\frac{4 + \sqrt{4 (3)}}{4 - \sqrt{12}}$

Step 1.1.2

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

$\frac{4 + \sqrt{2^{2} \cdot 3}}{4 - \sqrt{12}}$

Step 1.2

Pull terms out from under the radical.

$\frac{4 + 2 \sqrt{3}}{4 - \sqrt{12}}$

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

Factor $4$ out of $12$ .

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

Step 2.1.2

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

$\frac{4 + 2 \sqrt{3}}{4 - \sqrt{2^{2} \cdot 3}}$

Step 2.2

Pull terms out from under the radical.

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

Step 2.3

Multiply $2$ by $- 1$ .

$\frac{4 + 2 \sqrt{3}}{4 - 2 \sqrt{3}}$

Step 3

Cancel the common factor of $4 + 2 \sqrt{3}$ and $4 - 2 \sqrt{3}$ .

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Step 3.1

Factor $2$ out of $4$ .

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

Step 3.2

Factor $2$ out of $2 \sqrt{3}$ .

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

Step 3.3

Factor $2$ out of $2 (2) + 2 (\sqrt{3})$ .

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

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor $2$ out of $4$ .

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

Step 3.4.2

Factor $2$ out of $- 2 \sqrt{3}$ .

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

Step 3.4.3

Factor $2$ out of $2 (2) + 2 (- \sqrt{3})$ .

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

Step 3.4.4

Cancel the common factor.

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

Step 3.4.5

Rewrite the expression.

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

Step 4

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

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

Step 5

Combine fractions.

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Step 5.1

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

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

Step 5.2

Expand the denominator using the FOIL method.

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

Step 5.3

Simplify.

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

Step 5.4

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

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

Step 6

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

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $2$ by $2$ .

$4 + 2 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}$

Step 7.1.2

Move $2$ to the left of $\sqrt{3}$ .

$4 + 2 \sqrt{3} + 2 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}$

Step 7.1.3

Combine using the product rule for radicals.

$4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3 \cdot 3}$

Step 7.1.4

Multiply $3$ by $3$ .

$4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{9}$

Step 7.1.5

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

$4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3^{2}}$

Step 7.1.6

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

$4 + 2 \sqrt{3} + 2 \sqrt{3} + 3$

Step 7.2

Add $4$ and $3$ .

$7 + 2 \sqrt{3} + 2 \sqrt{3}$

Step 7.3

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

$7 + 4 \sqrt{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$7 + 4 \sqrt{3}$

Decimal Form:

$13.92820323 \dots$