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

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

Step 1

Simplify the denominator.

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

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Step 4

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 5

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $2$ by $2$ .

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $6$ .

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

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

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

Combine using the product rule for radicals.

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

Multiply $6$ by $2$ .

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

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

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $6$ .

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

Subtract $6$ from $4$ .

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

Add $- 2 \sqrt{3}$ and $4 \sqrt{3}$ .

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

Step 6

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

Factor $2$ out of $- 2$ .

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

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

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 1 + \sqrt{3}$ by $1$ .

$- 1 + \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 1 + \sqrt{3}$

Decimal Form:

$0.73205080 \dots$

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

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