Solve frac{3+sqrt{2}}{5-sqrt{5}}

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

Evaluate

$$\frac{\sqrt{10}+3\sqrt{5}+5\sqrt{2}+15}{20}\approx 1.59707747$$

Solution Steps

Rationalize the denominator of \frac{3+\sqrt{2}}{5-\sqrt{5}} by multiplying numerator and denominator by 5+\sqrt{5}.

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

Consider \left(5-\sqrt{5}\right)\left(5+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 5. Square \sqrt{5}.

$$\frac{\left(3+\sqrt{2}\right)\left(5+\sqrt{5}\right)}{25-5}$$

Subtract 5 from 25 to get 20.

$$\frac{\left(3+\sqrt{2}\right)\left(5+\sqrt{5}\right)}{20}$$

Apply the distributive property by multiplying each term of 3+\sqrt{2} by each term of 5+\sqrt{5}.

$$\frac{15+3\sqrt{5}+5\sqrt{2}+\sqrt{2}\sqrt{5}}{20}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{15+3\sqrt{5}+5\sqrt{2}+\sqrt{10}}{20}$$