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

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

Evaluate

$$\frac{\sqrt{10}+3\sqrt{5}-\sqrt{6}-3\sqrt{3}}{2}\approx 1.112419714$$

Solution Steps

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

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

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}.

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

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

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

Subtract 3 from 5 to get 2.

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

Apply the distributive property by multiplying each term of 3+\sqrt{2} by each term of \sqrt{5}-\sqrt{3}.

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

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

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

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

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