Simplify (4+ square root of 12)/(4- square root of 12)

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

Simplify the numerator.

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Simplify the denominator.

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

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

Factor $2$ out of $4$ .

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

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

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

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

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $3$ .

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

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

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

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

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

Add $4$ and $3$ .

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

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

$7 + 4 \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$7 + 4 \sqrt{3}$

Decimal Form:

$13.92820323 \dots$

Simplify (4+ square root of 12)/(4- square root of 12)

Related Questions

Support LCMGCF.com by disabling your adblocker.