Simplify (2- square root of 8)/(4+ square root of 12)

$\frac{2 - \sqrt{8}}{4 + \sqrt{12}}$

Step 1

Simplify the numerator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$\frac{2 - \sqrt{4 (2)}}{4 + \sqrt{12}}$

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

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

Pull terms out from under the radical.

$\frac{2 - (2 \sqrt{2})}{4 + \sqrt{12}}$

Multiply $2$ by $- 1$ .

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

Step 2

Simplify the denominator.

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Step 3

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

Factor $2$ out of $2$ .

$\frac{2 \cdot 1 - 2 \sqrt{2}}{4 + 2 \sqrt{3}}$

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

$\frac{2 \cdot 1 + 2 (- \sqrt{2})}{4 + 2 \sqrt{3}}$

Factor $2$ out of $2 (1) + 2 (- \sqrt{2})$ .

$\frac{2 (1 - \sqrt{2})}{4 + 2 \sqrt{3}}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1 - \sqrt{2}}{2 + \sqrt{3}}$

Step 4

Multiply $\frac{1 - \sqrt{2}}{2 + \sqrt{3}}$ by $\frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ .

$\frac{1 - \sqrt{2}}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}}$

Step 5

Combine fractions.

Multiply $\frac{1 - \sqrt{2}}{2 + \sqrt{3}}$ by $\frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ .

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

Expand the denominator using the FOIL method.

$\frac{(1 - \sqrt{2}) (2 - \sqrt{3})}{4 - 2 \sqrt{3} + \sqrt{3} \cdot 2 - {\sqrt{3}}^{2}}$

Simplify.

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

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

$(1 - \sqrt{2}) (2 - \sqrt{3})$

Step 6

Expand $(1 - \sqrt{2}) (2 - \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$1 (2 - \sqrt{3}) - \sqrt{2} (2 - \sqrt{3})$

Apply the distributive property.

$1 \cdot 2 + 1 (- \sqrt{3}) - \sqrt{2} (2 - \sqrt{3})$

Apply the distributive property.

$1 \cdot 2 + 1 (- \sqrt{3}) - \sqrt{2} \cdot 2 - \sqrt{2} (- \sqrt{3})$

Step 7

Simplify each term.

Multiply $2$ by $1$ .

$2 + 1 (- \sqrt{3}) - \sqrt{2} \cdot 2 - \sqrt{2} (- \sqrt{3})$

Multiply $- \sqrt{3}$ by $1$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- \sqrt{2} (- \sqrt{3})$ .

Multiply $- 1$ by $- 1$ .

$2 - \sqrt{3} - 2 \sqrt{2} + 1 \sqrt{2} \sqrt{3}$

Multiply $\sqrt{2}$ by $1$ .

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.11098818 \dots$

Simplify (2- square root of 8)/(4+ square root of 12)

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