Simplify ( square root of 2- square root of 6)/( square root of 6- square root of 8)

$\frac{\sqrt{2} - \sqrt{6}}{\sqrt{6} - \sqrt{8}}$

Simplify the denominator.

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

Factor $4$ out of $8$ .

$\frac{\sqrt{2} - \sqrt{6}}{\sqrt{6} - \sqrt{4 (2)}}$

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

$\frac{\sqrt{2} - \sqrt{6}}{\sqrt{6} - \sqrt{2^{2} \cdot 2}}$

Pull terms out from under the radical.

$\frac{\sqrt{2} - \sqrt{6}}{\sqrt{6} - (2 \sqrt{2})}$

Multiply $2$ by $- 1$ .

$\frac{\sqrt{2} - \sqrt{6}}{\sqrt{6} - 2 \sqrt{2}}$

Multiply $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{6} - 2 \sqrt{2}}$ by $\frac{\sqrt{6} + 2 \sqrt{2}}{\sqrt{6} + 2 \sqrt{2}}$ .

$\frac{\sqrt{2} - \sqrt{6}}{\sqrt{6} - 2 \sqrt{2}} \cdot \frac{\sqrt{6} + 2 \sqrt{2}}{\sqrt{6} + 2 \sqrt{2}}$

Combine fractions.

Multiply $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{6} - 2 \sqrt{2}}$ by $\frac{\sqrt{6} + 2 \sqrt{2}}{\sqrt{6} + 2 \sqrt{2}}$ .

$\frac{(\sqrt{2} - \sqrt{6}) (\sqrt{6} + 2 \sqrt{2})}{(\sqrt{6} - 2 \sqrt{2}) (\sqrt{6} + 2 \sqrt{2})}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{2} - \sqrt{6}) (\sqrt{6} + 2 \sqrt{2})}{{\sqrt{6}}^{2} + 2 \sqrt{12} - 2 \sqrt{12} - 4 {\sqrt{2}}^{2}}$

Simplify.

$\frac{(\sqrt{2} - \sqrt{6}) (\sqrt{6} + 2 \sqrt{2})}{- 2}$

Expand $(\sqrt{2} - \sqrt{6}) (\sqrt{6} + 2 \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{2} (\sqrt{6} + 2 \sqrt{2}) - \sqrt{6} (\sqrt{6} + 2 \sqrt{2})}{- 2}$

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{6} + \sqrt{2} (2 \sqrt{2}) - \sqrt{6} (\sqrt{6} + 2 \sqrt{2})}{- 2}$

Apply the distributive property.

$\frac{\sqrt{2} \sqrt{6} + \sqrt{2} (2 \sqrt{2}) - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 6} + \sqrt{2} (2 \sqrt{2}) - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Multiply $2$ by $6$ .

$\frac{\sqrt{12} + \sqrt{2} (2 \sqrt{2}) - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

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

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)} + \sqrt{2} (2 \sqrt{2}) - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

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

$\frac{\sqrt{2^{2} \cdot 3} + \sqrt{2} (2 \sqrt{2}) - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Pull terms out from under the radical.

$\frac{2 \sqrt{3} + \sqrt{2} (2 \sqrt{2}) - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

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

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 \sqrt{3} + 2 ({\sqrt{2}}^{1} \sqrt{2}) - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 \sqrt{3} + 2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt{3} + 2 {\sqrt{2}}^{1 + 1} - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Add $1$ and $1$ .

$\frac{2 \sqrt{3} + 2 {\sqrt{2}}^{2} - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{2 \sqrt{3} + 2 {(2^{\frac{1}{2}})}^{2} - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \sqrt{3} + 2 \cdot 2^{\frac{1}{2} \cdot 2} - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 \sqrt{3} + 2 \cdot 2^{\frac{2}{2}} - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{3} + 2 \cdot 2^{\frac{2}{2}} - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Rewrite the expression.

$\frac{2 \sqrt{3} + 2 \cdot 2^{1} - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Evaluate the exponent.

$\frac{2 \sqrt{3} + 2 \cdot 2 - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Multiply $2$ by $2$ .

$\frac{2 \sqrt{3} + 4 - \sqrt{6} \sqrt{6} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Multiply $- \sqrt{6} \sqrt{6}$ .

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{2 \sqrt{3} + 4 - ({\sqrt{6}}^{1} \sqrt{6}) - \sqrt{6} (2 \sqrt{2})}{- 2}$

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{2 \sqrt{3} + 4 - ({\sqrt{6}}^{1} {\sqrt{6}}^{1}) - \sqrt{6} (2 \sqrt{2})}{- 2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt{3} + 4 - {\sqrt{6}}^{1 + 1} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Add $1$ and $1$ .

$\frac{2 \sqrt{3} + 4 - {\sqrt{6}}^{2} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{2 \sqrt{3} + 4 - {(6^{\frac{1}{2}})}^{2} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \sqrt{3} + 4 - 6^{\frac{1}{2} \cdot 2} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 \sqrt{3} + 4 - 6^{\frac{2}{2}} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{3} + 4 - 6^{\frac{2}{2}} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Rewrite the expression.

$\frac{2 \sqrt{3} + 4 - 6^{1} - \sqrt{6} (2 \sqrt{2})}{- 2}$

Evaluate the exponent.

$\frac{2 \sqrt{3} + 4 - 1 \cdot 6 - \sqrt{6} (2 \sqrt{2})}{- 2}$

Multiply $- 1$ by $6$ .

$\frac{2 \sqrt{3} + 4 - 6 - \sqrt{6} (2 \sqrt{2})}{- 2}$

Multiply $- \sqrt{6} (2 \sqrt{2})$ .

Multiply $2$ by $- 1$ .

$\frac{2 \sqrt{3} + 4 - 6 - 2 \sqrt{6} \sqrt{2}}{- 2}$

Combine using the product rule for radicals.

$\frac{2 \sqrt{3} + 4 - 6 - 2 \sqrt{2 \cdot 6}}{- 2}$

Multiply $2$ by $6$ .

$\frac{2 \sqrt{3} + 4 - 6 - 2 \sqrt{12}}{- 2}$

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

Factor $4$ out of $12$ .

$\frac{2 \sqrt{3} + 4 - 6 - 2 \sqrt{4 (3)}}{- 2}$

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

$\frac{2 \sqrt{3} + 4 - 6 - 2 \sqrt{2^{2} \cdot 3}}{- 2}$

Pull terms out from under the radical.

$\frac{2 \sqrt{3} + 4 - 6 - 2 (2 \sqrt{3})}{- 2}$

Multiply $2$ by $- 2$ .

$\frac{2 \sqrt{3} + 4 - 6 - 4 \sqrt{3}}{- 2}$

Subtract $4 \sqrt{3}$ from $2 \sqrt{3}$ .

$\frac{4 - 6 - 2 \sqrt{3}}{- 2}$

Subtract $6$ from $4$ .

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

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

Factor $2$ out of $- 2$ .

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

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

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

Factor $2$ out of $2 (- 1) + 2 (- \sqrt{3})$ .

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

Move the negative one from the denominator of $\frac{- 1 - \sqrt{3}}{- 1}$ .

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

Rewrite $- 1 \cdot (- 1 - \sqrt{3})$ as $- (- 1 - \sqrt{3})$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

$1 - - \sqrt{3}$

Multiply $- - \sqrt{3}$ .

Multiply $- 1$ by $- 1$ .

$1 + 1 \sqrt{3}$

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

$1 + \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$1 + \sqrt{3}$

Decimal Form:

$2.73205080 \dots$

Simplify ( square root of 2- square root of 6)/( square root of 6- square root of 8)

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