Solve frac{sqrt{2}}{4+sqrt{3}}

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

Evaluate

$$-\frac{\sqrt{2}\left(\sqrt{3}-4\right)}{13}\approx 0.246720347$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{2}}{4+\sqrt{3}} by multiplying numerator and denominator by 4-\sqrt{3}.

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

Consider \left(4+\sqrt{3}\right)\left(4-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 4. Square \sqrt{3}.

$$\frac{\sqrt{2}\left(4-\sqrt{3}\right)}{16-3}$$

Subtract 3 from 16 to get 13.

$$\frac{\sqrt{2}\left(4-\sqrt{3}\right)}{13}$$

Use the distributive property to multiply \sqrt{2} by 4-\sqrt{3}.

$$\frac{4\sqrt{2}-\sqrt{2}\sqrt{3}}{13}$$

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

$$\frac{4\sqrt{2}-\sqrt{6}}{13}$$