Simplify 2/(1+ square root of 5)

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

$\frac{2 (1 - \sqrt{5})}{- 4}$

Step 5

Cancel the common factors.

Factor $2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Move the negative in front of the fraction.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.61803398 \dots$