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

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

Evaluate

$$\frac{-\sqrt{10}-5}{3}\approx -2.72075922$$

Solution Steps

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

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

Consider \left(\sqrt{2}-\sqrt{5}\right)\left(\sqrt{2}+\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}.

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

Square \sqrt{2}. Square \sqrt{5}.

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

Subtract 5 from 2 to get -3.

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

Use the distributive property to multiply \sqrt{5} by \sqrt{2}+\sqrt{5}.

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

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

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

The square of \sqrt{5} is 5.

$$\frac{\sqrt{10}+5}{-3}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{10}-5}{3}$$