Solve frac{2+sqrt{6}}{sqrt{5}+sqrt{3}}

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

Evaluate

$$\frac{\sqrt{30}}{2}-\frac{3\sqrt{2}}{2}+\sqrt{5}-\sqrt{3}\approx 1.121309614$$

Solution Steps

Rationalize the denominator of \frac{2+\sqrt{6}}{\sqrt{5}+\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}-\sqrt{3}.

$$\frac{\left(2+\sqrt{6}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}$$

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}.

$$\frac{\left(2+\sqrt{6}\right)\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{\left(2+\sqrt{6}\right)\left(\sqrt{5}-\sqrt{3}\right)}{5-3}$$

Subtract 3 from 5 to get 2.

$$\frac{\left(2+\sqrt{6}\right)\left(\sqrt{5}-\sqrt{3}\right)}{2}$$

Apply the distributive property by multiplying each term of 2+\sqrt{6} by each term of \sqrt{5}-\sqrt{3}.

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

To multiply \sqrt{6} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{2\sqrt{5}-2\sqrt{3}+\sqrt{30}-\sqrt{6}\sqrt{3}}{2}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{2\sqrt{5}-2\sqrt{3}+\sqrt{30}-\sqrt{3}\sqrt{2}\sqrt{3}}{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{2\sqrt{5}-2\sqrt{3}+\sqrt{30}-3\sqrt{2}}{2}$$