Solve 1/sqrt{5+sqrt{3}}+1/sqrt{5-sqrt{3}}

$$\frac{ 1 }{ \sqrt{ 5 } + \sqrt{ 3 } } + \frac{ 1 }{ \sqrt{ 5 } - \sqrt{ 3 } }$$

Evaluate

$$\sqrt{5}\approx 2.236067977$$

Short Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{5}+\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}-\sqrt{3}.

$$\frac{\sqrt{5}-\sqrt{3}}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}+\frac{1}{\sqrt{5}-\sqrt{3}}$$

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}.

$$\frac{\sqrt{5}-\sqrt{3}}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}+\frac{1}{\sqrt{5}-\sqrt{3}}$$

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

$$\frac{\sqrt{5}-\sqrt{3}}{5-3}+\frac{1}{\sqrt{5}-\sqrt{3}}$$

Subtract 3 from 5 to get 2.

$$\frac{\sqrt{5}-\sqrt{3}}{2}+\frac{1}{\sqrt{5}-\sqrt{3}}$$

Rationalize the denominator of \frac{1}{\sqrt{5}-\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}+\sqrt{3}.

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

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}.

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

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

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

Subtract 3 from 5 to get 2.

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

Since \frac{\sqrt{5}-\sqrt{3}}{2} and \frac{\sqrt{5}+\sqrt{3}}{2} have the same denominator, add them by adding their numerators.

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

Do the calculations in \sqrt{5}-\sqrt{3}+\sqrt{5}+\sqrt{3}.

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

Cancel out 2 and 2.

$$\sqrt{5}$$