Algebra Examples

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

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 $12$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{2 \sqrt{3}}{4 - \sqrt{3}}$ by $\frac{4 + \sqrt{3}}{4 + \sqrt{3}}$ .

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{2 \sqrt{3}}{4 - \sqrt{3}}$ by $\frac{4 + \sqrt{3}}{4 + \sqrt{3}}$ .

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

Step 3.2

Expand the denominator using the FOIL method.

$\frac{2 \sqrt{3} (4 + \sqrt{3})}{16 + 4 \sqrt{3} - 4 \sqrt{3} - {\sqrt{3}}^{2}}$

Step 3.3

Simplify.

$\frac{2 \sqrt{3} (4 + \sqrt{3})}{13}$

Step 3.4

Group $4 + \sqrt{3}$ and $\sqrt{3}$ together.

$\frac{2 ((4 + \sqrt{3}) \sqrt{3})}{13}$

Step 3.5

Apply the distributive property.

$\frac{2 (4 \sqrt{3} + \sqrt{3} \sqrt{3})}{13}$

Step 3.6

Combine using the product rule for radicals.

$\frac{2 (4 \sqrt{3} + \sqrt{3 \cdot 3})}{13}$

Step 4

Simplify each term.

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Step 4.1

Multiply $3$ by $3$ .

$\frac{2 (4 \sqrt{3} + \sqrt{9})}{13}$

Step 4.2

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

$\frac{2 (4 \sqrt{3} + \sqrt{3^{2}})}{13}$

Step 4.3

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

$\frac{2 (4 \sqrt{3} + 3)}{13}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2 (4 \sqrt{3} + 3)}{13}$

Decimal Form:

$1.52741588 \dots$