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

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

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.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Step 3

Simplify the numerator.

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

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

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

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

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

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

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.

Apply the distributive property.

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

Apply the distributive property.

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

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.

Simplify each term.

Multiply $2$ by $2$ .

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $- 1$ .

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

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

Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Add $4$ and $6$ .

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

Subtract $2 \sqrt{6}$ from $- 2 \sqrt{6}$ .

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

Step 7

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

Factor $2$ out of $10$ .

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

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

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

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

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

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.

Multiply $- 1$ by $5$ .

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

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:

$- 0.10102051 \dots$

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

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