Solve frac{sqrt{3}}{4+sqrt{10}}

$$\frac { \sqrt { 3 } } { 4 + \sqrt { 10 } }$$

Evaluate

$$-\frac{\sqrt{3}\left(\sqrt{10}-4\right)}{6}\approx 0.241829609$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}}{4+\sqrt{10}} by multiplying numerator and denominator by 4-\sqrt{10}.

$$\frac{\sqrt{3}\left(4-\sqrt{10}\right)}{\left(4+\sqrt{10}\right)\left(4-\sqrt{10}\right)}$$

Consider \left(4+\sqrt{10}\right)\left(4-\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}.

$$\frac{\sqrt{3}\left(4-\sqrt{10}\right)}{4^{2}-\left(\sqrt{10}\right)^{2}}$$

Square 4. Square \sqrt{10}.

$$\frac{\sqrt{3}\left(4-\sqrt{10}\right)}{16-10}$$

Subtract 10 from 16 to get 6.

$$\frac{\sqrt{3}\left(4-\sqrt{10}\right)}{6}$$

Use the distributive property to multiply \sqrt{3} by 4-\sqrt{10}.

$$\frac{4\sqrt{3}-\sqrt{3}\sqrt{10}}{6}$$

To multiply \sqrt{3} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{4\sqrt{3}-\sqrt{30}}{6}$$