Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

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

Multiply $2$ by $2$ .

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

Step 6.1.2

Move $2$ to the left of $\sqrt{6}$ .

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

Step 6.1.3

Combine using the product rule for radicals.

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

Step 6.1.4

Multiply $6$ by $6$ .

$\frac{4 + 2 \sqrt{6} + 2 \sqrt{6} + \sqrt{36}}{- 2}$

Step 6.1.5

Rewrite $36$ as $6^{2}$ .

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

Step 6.1.6

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

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

Step 6.2

Add $4$ and $6$ .

$\frac{10 + 2 \sqrt{6} + 2 \sqrt{6}}{- 2}$

Step 6.3

Add $2 \sqrt{6}$ and $2 \sqrt{6}$ .

$\frac{10 + 4 \sqrt{6}}{- 2}$

Step 7

Cancel the common factor of $10 + 4 \sqrt{6}$ and $- 2$ .

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Step 7.1

Factor $2$ out of $10$ .

$\frac{2 (5) + 4 \sqrt{6}}{- 2}$

Step 7.2

Factor $2$ out of $4 \sqrt{6}$ .

$\frac{2 (5) + 2 (2 \sqrt{6})}{- 2}$

Step 7.3

Factor $2$ out of $2 (5) + 2 (2 \sqrt{6})$ .

$\frac{2 (5 + 2 \sqrt{6})}{- 2}$

Step 7.4

Move the negative one from the denominator of $\frac{5 + 2 \sqrt{6}}{- 1}$ .

$- 1 \cdot (5 + 2 \sqrt{6})$

Step 8

Rewrite $- 1 \cdot (5 + 2 \sqrt{6})$ as $- (5 + 2 \sqrt{6})$ .

$- (5 + 2 \sqrt{6})$

Step 9

Apply the distributive property.

$- 1 \cdot 5 - (2 \sqrt{6})$

Step 10

Multiply.

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Step 10.1

Multiply $- 1$ by $5$ .

$- 5 - (2 \sqrt{6})$

Step 10.2

Multiply $2$ by $- 1$ .

$- 5 - 2 \sqrt{6}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- 5 - 2 \sqrt{6}$

Decimal Form:

$- 9.89897948 \dots$