Evaluate (2- square root of 5)/(6+2 square root of 5)

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

Step 1

Multiply $\frac{2 - \sqrt{5}}{6 + 2 \sqrt{5}}$ by $\frac{6 - 2 \sqrt{5}}{6 - 2 \sqrt{5}}$ .

$\frac{2 - \sqrt{5}}{6 + 2 \sqrt{5}} \cdot \frac{6 - 2 \sqrt{5}}{6 - 2 \sqrt{5}}$

Step 2

Simplify terms.

Multiply $\frac{2 - \sqrt{5}}{6 + 2 \sqrt{5}}$ by $\frac{6 - 2 \sqrt{5}}{6 - 2 \sqrt{5}}$ .

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

Expand the denominator using the FOIL method.

$\frac{(2 - \sqrt{5}) (6 - 2 \sqrt{5})}{36 - 12 \sqrt{5} + 12 \sqrt{5} - 4 {\sqrt{5}}^{2}}$

Simplify.

$\frac{(2 - \sqrt{5}) (6 - 2 \sqrt{5})}{16}$

Cancel the common factor of $6 - 2 \sqrt{5}$ and $16$ .

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

$\frac{2 ((2 - \sqrt{5}) (3 - \sqrt{5}))}{16}$

Cancel the common factors.

Factor $2$ out of $16$ .

$\frac{2 ((2 - \sqrt{5}) (3 - \sqrt{5}))}{2 (8)}$

Cancel the common factor.

$\frac{2 ((2 - \sqrt{5}) (3 - \sqrt{5}))}{2 \cdot 8}$

Rewrite the expression.

$\frac{(2 - \sqrt{5}) (3 - \sqrt{5})}{8}$

Step 3

Expand $(2 - \sqrt{5}) (3 - \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

$\frac{2 (3 - \sqrt{5}) - \sqrt{5} (3 - \sqrt{5})}{8}$

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- \sqrt{5}) - \sqrt{5} (3 - \sqrt{5})}{8}$

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- \sqrt{5}) - \sqrt{5} \cdot 3 - \sqrt{5} (- \sqrt{5})}{8}$

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $3$ .

$\frac{6 + 2 (- \sqrt{5}) - \sqrt{5} \cdot 3 - \sqrt{5} (- \sqrt{5})}{8}$

Multiply $- 1$ by $2$ .

$\frac{6 - 2 \sqrt{5} - \sqrt{5} \cdot 3 - \sqrt{5} (- \sqrt{5})}{8}$

Multiply $3$ by $- 1$ .

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} - \sqrt{5} (- \sqrt{5})}{8}$

Multiply $- \sqrt{5} (- \sqrt{5})$ .

Multiply $- 1$ by $- 1$ .

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + 1 \sqrt{5} \sqrt{5}}{8}$

Multiply $\sqrt{5}$ by $1$ .

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + \sqrt{5} \sqrt{5}}{8}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + {\sqrt{5}}^{1} \sqrt{5}}{8}$

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + {\sqrt{5}}^{1} {\sqrt{5}}^{1}}{8}$

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

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + {\sqrt{5}}^{1 + 1}}{8}$

Add $1$ and $1$ .

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + {\sqrt{5}}^{2}}{8}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + {(5^{\frac{1}{2}})}^{2}}{8}$

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

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + 5^{\frac{1}{2} \cdot 2}}{8}$

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

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + 5^{\frac{2}{2}}}{8}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + 5^{\frac{2}{2}}}{8}$

Rewrite the expression.

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + 5^{1}}{8}$

Evaluate the exponent.

$\frac{6 - 2 \sqrt{5} - 3 \sqrt{5} + 5}{8}$

Add $6$ and $5$ .

$\frac{11 - 2 \sqrt{5} - 3 \sqrt{5}}{8}$

Subtract $3 \sqrt{5}$ from $- 2 \sqrt{5}$ .

$\frac{11 - 5 \sqrt{5}}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{11 - 5 \sqrt{5}}{8}$

Decimal Form:

$- 0.02254248 \dots$

Evaluate (2- square root of 5)/(6+2 square root of 5)

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