Solve frac{2sqrt{6}}{sqrt{7}+sqrt{5}}

$$\frac { 2 \sqrt { 6 } } { \sqrt { 7 } + \sqrt { 5 } }$$

Evaluate

$$\sqrt{42}-\sqrt{30}\approx 1.003515123$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{6}}{\sqrt{7}+\sqrt{5}} by multiplying numerator and denominator by \sqrt{7}-\sqrt{5}.

$$\frac{2\sqrt{6}\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}$$

Consider \left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\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}.

$$\frac{2\sqrt{6}\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

Square \sqrt{7}. Square \sqrt{5}.

$$\frac{2\sqrt{6}\left(\sqrt{7}-\sqrt{5}\right)}{7-5}$$

Subtract 5 from 7 to get 2.

$$\frac{2\sqrt{6}\left(\sqrt{7}-\sqrt{5}\right)}{2}$$

Use the distributive property to multiply 2\sqrt{6} by \sqrt{7}-\sqrt{5}.

$$\frac{2\sqrt{6}\sqrt{7}-2\sqrt{6}\sqrt{5}}{2}$$

To multiply \sqrt{6} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{2\sqrt{42}-2\sqrt{6}\sqrt{5}}{2}$$

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

$$\frac{2\sqrt{42}-2\sqrt{30}}{2}$$

Divide each term of 2\sqrt{42}-2\sqrt{30} by 2 to get \sqrt{42}-\sqrt{30}.

$$\sqrt{42}-\sqrt{30}$$