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

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

Evaluate

$$-\sqrt{3}\approx -1.732050808$$

Short Solution Steps

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

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

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}.

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

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

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

Subtract 5 from 3 to get -2.

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

Multiply both numerator and denominator by -1.

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

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

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

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}.

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

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

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

Subtract 5 from 3 to get -2.

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

Multiply both numerator and denominator by -1.

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

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

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

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

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

Cancel out 2 and 2.

$$-\sqrt{3}$$