Solve frac{sqrt{7}}{sqrt{10}-sqrt{2}}

$$\frac { \sqrt { 7 } } { \sqrt { 10 } - \sqrt { 2 } }$$

Evaluate

$$\frac{\sqrt{14}+\sqrt{70}}{8}\approx 1.513532207$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{10}-\sqrt{2}} by multiplying numerator and denominator by \sqrt{10}+\sqrt{2}.

$$\frac{\sqrt{7}\left(\sqrt{10}+\sqrt{2}\right)}{\left(\sqrt{10}-\sqrt{2}\right)\left(\sqrt{10}+\sqrt{2}\right)}$$

Consider \left(\sqrt{10}-\sqrt{2}\right)\left(\sqrt{10}+\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{7}\left(\sqrt{10}+\sqrt{2}\right)}{\left(\sqrt{10}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square \sqrt{10}. Square \sqrt{2}.

$$\frac{\sqrt{7}\left(\sqrt{10}+\sqrt{2}\right)}{10-2}$$

Subtract 2 from 10 to get 8.

$$\frac{\sqrt{7}\left(\sqrt{10}+\sqrt{2}\right)}{8}$$

Use the distributive property to multiply \sqrt{7} by \sqrt{10}+\sqrt{2}.

$$\frac{\sqrt{7}\sqrt{10}+\sqrt{7}\sqrt{2}}{8}$$

To multiply \sqrt{7} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{\sqrt{70}+\sqrt{7}\sqrt{2}}{8}$$

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

$$\frac{\sqrt{70}+\sqrt{14}}{8}$$