Simplify ( square root of 3)/( square root of 2+ square root of 5)

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{3} (\sqrt{2} - \sqrt{5})}{{\sqrt{2}}^{2} - \sqrt{10} + \sqrt{10} - {\sqrt{5}}^{2}}$

Step 4

Simplify.

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

Step 5

Apply the distributive property.

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

Step 6

Combine using the product rule for radicals.

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

Step 7

Multiply $\sqrt{3} (- \sqrt{5})$ .

Combine using the product rule for radicals.

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

Multiply $3$ by $5$ .

$\frac{\sqrt{3 \cdot 2} - \sqrt{15}}{- 3}$

Step 8

Multiply $3$ by $2$ .

$\frac{\sqrt{6} - \sqrt{15}}{- 3}$

Step 9

Move the negative in front of the fraction.

$- \frac{\sqrt{6} - \sqrt{15}}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{6} - \sqrt{15}}{3}$

Decimal Form:

$0.47449786 \dots$