Simplify the Radical Expression 3/(2+ square root of 5)

$\frac{3}{2 + \sqrt{5}}$

Multiply $\frac{3}{2 + \sqrt{5}}$ by $\frac{2 - \sqrt{5}}{2 - \sqrt{5}}$ .

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

Multiply $\frac{3}{2 + \sqrt{5}}$ by $\frac{2 - \sqrt{5}}{2 - \sqrt{5}}$ .

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

Expand the denominator using the FOIL method.

$\frac{3 (2 - \sqrt{5})}{4 - 2 \sqrt{5} + \sqrt{5} \cdot 2 - {\sqrt{5}}^{2}}$

Simplify.

$\frac{3 (2 - \sqrt{5})}{- 1}$

Move the negative one from the denominator of $\frac{3 (2 - \sqrt{5})}{- 1}$ .

$- 1 \cdot (3 (2 - \sqrt{5}))$

Rewrite $- 1 \cdot (3 (2 - \sqrt{5}))$ as $- (3 (2 - \sqrt{5}))$ .

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

Apply the distributive property.

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

Multiply $3$ by $2$ .

$- (6 + 3 (- \sqrt{5}))$

Multiply $- 1$ by $3$ .

$- (6 - 3 \sqrt{5})$

Apply the distributive property.

$- 1 \cdot 6 - (- 3 \sqrt{5})$

Multiply $- 1$ by $6$ .

$- 6 - (- 3 \sqrt{5})$

Multiply $- 3$ by $- 1$ .

$- 6 + 3 \sqrt{5}$

The result can be shown in multiple forms.

Exact Form:

$- 6 + 3 \sqrt{5}$

Decimal Form:

$0.70820393 \dots$