Solve frac{sqrt{2}}{sqrt{3}-sqrt{7}}

$$\frac { \sqrt { 2 } } { \sqrt { 3 } - \sqrt { 7 } }$$

Evaluate

$$\frac{-\sqrt{6}-\sqrt{14}}{4}\approx -1.547786782$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{3}-\sqrt{7}} by multiplying numerator and denominator by \sqrt{3}+\sqrt{7}.

$$\frac{\sqrt{2}\left(\sqrt{3}+\sqrt{7}\right)}{\left(\sqrt{3}-\sqrt{7}\right)\left(\sqrt{3}+\sqrt{7}\right)}$$

Consider \left(\sqrt{3}-\sqrt{7}\right)\left(\sqrt{3}+\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{2}\left(\sqrt{3}+\sqrt{7}\right)}{\left(\sqrt{3}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

Square \sqrt{3}. Square \sqrt{7}.

$$\frac{\sqrt{2}\left(\sqrt{3}+\sqrt{7}\right)}{3-7}$$

Subtract 7 from 3 to get -4.

$$\frac{\sqrt{2}\left(\sqrt{3}+\sqrt{7}\right)}{-4}$$

Use the distributive property to multiply \sqrt{2} by \sqrt{3}+\sqrt{7}.

$$\frac{\sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{7}}{-4}$$

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

$$\frac{\sqrt{6}+\sqrt{2}\sqrt{7}}{-4}$$

To multiply \sqrt{2} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{6}+\sqrt{14}}{-4}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{6}-\sqrt{14}}{4}$$