Solve frac{sqrt{7}}{3-sqrt{10}}

$$\frac { \sqrt { 7 } } { 3 - \sqrt { 10 } }$$

Evaluate

$$-\sqrt{7}\left(\sqrt{10}+3\right)\approx -16.303854199$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{7}}{3-\sqrt{10}} by multiplying numerator and denominator by 3+\sqrt{10}.

$$\frac{\sqrt{7}\left(3+\sqrt{10}\right)}{\left(3-\sqrt{10}\right)\left(3+\sqrt{10}\right)}$$

Consider \left(3-\sqrt{10}\right)\left(3+\sqrt{10}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{7}\left(3+\sqrt{10}\right)}{3^{2}-\left(\sqrt{10}\right)^{2}}$$

Square 3. Square \sqrt{10}.

$$\frac{\sqrt{7}\left(3+\sqrt{10}\right)}{9-10}$$

Subtract 10 from 9 to get -1.

$$\frac{\sqrt{7}\left(3+\sqrt{10}\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-\sqrt{7}\left(3+\sqrt{10}\right)$$

Use the distributive property to multiply \sqrt{7} by 3+\sqrt{10}.

$$-\left(3\sqrt{7}+\sqrt{7}\sqrt{10}\right)$$

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

$$-\left(3\sqrt{7}+\sqrt{70}\right)$$

To find the opposite of 3\sqrt{7}+\sqrt{70}, find the opposite of each term.

$$-3\sqrt{7}-\sqrt{70}$$