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

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

Evaluate

$$-\left(\sqrt{30}+3\sqrt{2}\right)\approx -9.719866262$$

Solution Steps

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

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

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}.

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

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

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

Subtract 5 from 3 to get -2.

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

Use the distributive property to multiply 2\sqrt{6} by \sqrt{3}+\sqrt{5}.

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

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}.

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

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

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

Multiply 2 and 3 to get 6.

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

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

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

Divide each term of 6\sqrt{2}+2\sqrt{30} by -2 to get -3\sqrt{2}-\sqrt{30}.

$$-3\sqrt{2}-\sqrt{30}$$