Solve frac{1+sqrt{2}}{sqrt{5}+sqrt{3}}+frac{1-sqrt{2}}{sqrt{5}+sqrt{3}}

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

Evaluate

$$\sqrt{5}-\sqrt{3}\approx 0.50401717$$

Short Solution Steps

Rationalize the denominator of \frac{1+\sqrt{2}}{\sqrt{5}+\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}-\sqrt{3}.

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

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}.

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

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

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

Subtract 3 from 5 to get 2.

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

Rationalize the denominator of \frac{1-\sqrt{2}}{\sqrt{5}+\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}-\sqrt{3}.

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

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}.

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

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

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

Subtract 3 from 5 to get 2.

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

Since \frac{\left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2} and \frac{\left(1-\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(1+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right)+\left(1-\sqrt{2}\right)\left(\sqrt{5}-\sqrt{3}\right).

$$\frac{\sqrt{5}-\sqrt{3}+\sqrt{10}-\sqrt{6}+\sqrt{5}-\sqrt{3}-\sqrt{10}+\sqrt{6}}{2}$$

Do the calculations in \sqrt{5}-\sqrt{3}+\sqrt{10}-\sqrt{6}+\sqrt{5}-\sqrt{3}-\sqrt{10}+\sqrt{6}.

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

Divide each term of 2\sqrt{5}-2\sqrt{3} by 2 to get \sqrt{5}-\sqrt{3}.

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

Factor

$$\sqrt{5} - \sqrt{3} = 0.50401717$$