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

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

$\frac{5 (3 \sqrt{5} - \sqrt{2})}{9 {\sqrt{5}}^{2} - 3 \sqrt{10} + 3 \sqrt{10} - {\sqrt{2}}^{2}}$

Step 4

Simplify.

$\frac{5 (3 \sqrt{5} - \sqrt{2})}{43}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{5 (3 \sqrt{5} - \sqrt{2})}{43}$

Decimal Form:

$0.61558027 \dots$