Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $8$ .

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

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $2$ by $- 1$ .

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

Step 2

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

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

Step 3

Combine fractions.

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Step 3.1

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

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

Step 3.2

Expand the denominator using the FOIL method.

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

Step 3.3

Simplify.

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Combine using the product rule for radicals.

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

Step 5.1.2

Multiply $2$ by $6$ .

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

Step 5.1.3

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

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Step 5.1.3.1

Factor $4$ out of $12$ .

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

Step 5.1.3.2

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

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

Step 5.1.4

Pull terms out from under the radical.

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

Step 5.1.5

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

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Step 5.1.5.1

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

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

Step 5.1.5.2

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

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

Step 5.1.5.3

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

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

Step 5.1.5.4

Add $1$ and $1$ .

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

Step 5.1.6

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

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Step 5.1.6.1

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

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

Step 5.1.6.2

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

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

Step 5.1.6.3

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

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

Step 5.1.6.4

Cancel the common factor of $2$ .

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Step 5.1.6.4.1

Cancel the common factor.

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

Step 5.1.6.4.2

Rewrite the expression.

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

Step 5.1.6.5

Evaluate the exponent.

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

Step 5.1.7

Multiply $2$ by $2$ .

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

Step 5.1.8

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

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Step 5.1.8.1

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

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

Step 5.1.8.2

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

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

Step 5.1.8.3

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

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

Step 5.1.8.4

Add $1$ and $1$ .

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

Step 5.1.9

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

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Step 5.1.9.1

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

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

Step 5.1.9.2

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

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

Step 5.1.9.3

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

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

Step 5.1.9.4

Cancel the common factor of $2$ .

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Step 5.1.9.4.1

Cancel the common factor.

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

Step 5.1.9.4.2

Rewrite the expression.

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

Step 5.1.9.5

Evaluate the exponent.

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

Step 5.1.10

Multiply $- 1$ by $6$ .

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

Step 5.1.11

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

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Step 5.1.11.1

Multiply $2$ by $- 1$ .

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

Step 5.1.11.2

Combine using the product rule for radicals.

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

Step 5.1.11.3

Multiply $2$ by $6$ .

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

Step 5.1.12

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

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Step 5.1.12.1

Factor $4$ out of $12$ .

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

Step 5.1.12.2

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

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

Step 5.1.13

Pull terms out from under the radical.

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

Step 5.1.14

Multiply $2$ by $- 2$ .

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

Step 5.2

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

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

Step 5.3

Subtract $6$ from $4$ .

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

Step 6

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

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Step 6.1

Factor $2$ out of $- 2$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 7

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

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

Step 8

Apply the distributive property.

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

Step 9

Multiply $- 1$ by $- 1$ .

$1 - - \sqrt{3}$

Step 10

Multiply $- - \sqrt{3}$ .

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Step 10.1

Multiply $- 1$ by $- 1$ .

$1 + 1 \sqrt{3}$

Step 10.2

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

$1 + \sqrt{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$1 + \sqrt{3}$

Decimal Form:

$2.73205080 \dots$