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

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

Evaluate

$$\frac{-\sqrt{6}-\sqrt{10}}{2}\approx -2.805883701$$

Solution Steps

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

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

Consider \left(\sqrt{3}-\sqrt{5}\right)\left(\sqrt{3}+\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}.

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

Square \sqrt{3}. Square \sqrt{5}.

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

Subtract 5 from 3 to get -2.

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

Use the distributive property to multiply \sqrt{2} by \sqrt{3}+\sqrt{5}.

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

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{6}+\sqrt{2}\sqrt{5}}{-2}$$

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

$$\frac{\sqrt{6}+\sqrt{10}}{-2}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{6}-\sqrt{10}}{2}$$