Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

Raise $\sqrt{2} - \sqrt{6}$ to the power of $1$ .

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

Step 3.2

Raise $\sqrt{2} - \sqrt{6}$ to the power of $1$ .

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

Rewrite ${(\sqrt{2} - \sqrt{6})}^{2}$ as $(\sqrt{2} - \sqrt{6}) (\sqrt{2} - \sqrt{6})$ .

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Combine using the product rule for radicals.

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

Step 6.1.2

Multiply $2$ by $2$ .

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

Step 6.1.3

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

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

Step 6.1.4

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

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

Step 6.1.5

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

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Step 6.1.5.1

Combine using the product rule for radicals.

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

Step 6.1.5.2

Multiply $2$ by $6$ .

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

Step 6.1.6

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

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Step 6.1.6.1

Factor $4$ out of $12$ .

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

Step 6.1.6.2

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

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

Step 6.1.7

Pull terms out from under the radical.

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

Step 6.1.8

Multiply $2$ by $- 1$ .

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

Step 6.1.9

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

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Step 6.1.9.1

Combine using the product rule for radicals.

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

Step 6.1.9.2

Multiply $2$ by $6$ .

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

Step 6.1.10

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

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Step 6.1.10.1

Factor $4$ out of $12$ .

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

Step 6.1.10.2

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

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

Step 6.1.11

Pull terms out from under the radical.

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

Step 6.1.12

Multiply $2$ by $- 1$ .

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

Step 6.1.13

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

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Step 6.1.13.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.13.2

Multiply $\sqrt{6}$ by $1$ .

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

Step 6.1.13.3

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

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

Step 6.1.13.4

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

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

Step 6.1.13.5

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

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

Step 6.1.13.6

Add $1$ and $1$ .

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

Step 6.1.14

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

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Step 6.1.14.1

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

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

Step 6.1.14.2

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

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

Step 6.1.14.3

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

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

Step 6.1.14.4

Cancel the common factor of $2$ .

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Step 6.1.14.4.1

Cancel the common factor.

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

Step 6.1.14.4.2

Rewrite the expression.

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

Step 6.1.14.5

Evaluate the exponent.

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

Step 6.2

Add $2$ and $6$ .

$\frac{8 - 2 \sqrt{3} - 2 \sqrt{3}}{- 4}$

Step 6.3

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

$\frac{8 - 4 \sqrt{3}}{- 4}$

Step 7

Cancel the common factor of $8 - 4 \sqrt{3}$ and $- 4$ .

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Step 7.1

Factor $4$ out of $8$ .

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

Step 7.2

Factor $4$ out of $- 4 \sqrt{3}$ .

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

Step 7.3

Factor $4$ out of $4 (2) + 4 (- \sqrt{3})$ .

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

Step 7.4

Move the negative one from the denominator of $\frac{2 - \sqrt{3}}{- 1}$ .

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

Step 8

Rewrite $- 1 \cdot (2 - \sqrt{3})$ as $- (2 - \sqrt{3})$ .

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

Step 9

Apply the distributive property.

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

Step 10

Multiply $- 1$ by $2$ .

$- 2 - - \sqrt{3}$

Step 11

Multiply $- - \sqrt{3}$ .

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Step 11.1

Multiply $- 1$ by $- 1$ .

$- 2 + 1 \sqrt{3}$

Step 11.2

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

$- 2 + \sqrt{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt{3}$

Decimal Form:

$- 0.26794919 \dots$