Evaluate ( square root of 2)/(4+ square root of 5)

$\frac{\sqrt{2}}{4 + \sqrt{5}}$

Step 1

Multiply $\frac{\sqrt{2}}{4 + \sqrt{5}}$ by $\frac{4 - \sqrt{5}}{4 - \sqrt{5}}$ .

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

Step 2

Multiply $\frac{\sqrt{2}}{4 + \sqrt{5}}$ by $\frac{4 - \sqrt{5}}{4 - \sqrt{5}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{2} (4 - \sqrt{5})}{16 - 4 \sqrt{5} + \sqrt{5} \cdot 4 - {\sqrt{5}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{2} (4 - \sqrt{5})}{11}$

Step 5

Apply the distributive property.

$\frac{\sqrt{2} \cdot 4 + \sqrt{2} (- \sqrt{5})}{11}$

Step 6

Move $4$ to the left of $\sqrt{2}$ .

$\frac{4 \cdot \sqrt{2} + \sqrt{2} (- \sqrt{5})}{11}$

Step 7

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

Combine using the product rule for radicals.

$\frac{4 \cdot \sqrt{2} - \sqrt{2 \cdot 5}}{11}$

Multiply $2$ by $5$ .

$\frac{4 \cdot \sqrt{2} - \sqrt{10}}{11}$

$\frac{4 \sqrt{2} - \sqrt{10}}{11}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{4 \sqrt{2} - \sqrt{10}}{11}$

Decimal Form:

$0.22677968 \dots$