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

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

Evaluate

$$\frac{\sqrt{15}+5}{2}\approx 4.436491673$$

Solution Steps

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

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

Consider \left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Subtract 3 from 5 to get 2.

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

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

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

The square of \sqrt{5} is 5.

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

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

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