Algebra Examples

$\frac{1}{2 + \sqrt{5}}$

Step 1

Multiply $\frac{1}{2 + \sqrt{5}}$ by $\frac{2 - \sqrt{5}}{2 - \sqrt{5}}$ .

$\frac{1}{2 + \sqrt{5}} \cdot \frac{2 - \sqrt{5}}{2 - \sqrt{5}}$

Step 2

Multiply $\frac{1}{2 + \sqrt{5}}$ by $\frac{2 - \sqrt{5}}{2 - \sqrt{5}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{2 - \sqrt{5}}{4 - 2 \sqrt{5} + \sqrt{5} \cdot 2 - {\sqrt{5}}^{2}}$

Step 4

Simplify.

$\frac{2 - \sqrt{5}}{- 1}$

Step 5

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

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

Step 6

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

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

Step 7

Apply the distributive property.

$- 1 \cdot 2 - - \sqrt{5}$

Step 8

Multiply $- 1$ by $2$ .

$- 2 - - \sqrt{5}$

Step 9

Multiply $- - \sqrt{5}$ .

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Step 9.1

Multiply $- 1$ by $- 1$ .

$- 2 + 1 \sqrt{5}$

Step 9.2

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

$- 2 + \sqrt{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt{5}$

Decimal Form:

$0.23606797 \dots$