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

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

Evaluate

$$-\frac{\sqrt{3}\left(\sqrt{2}+4\right)}{14}\approx -0.669835212$$

Solution Steps

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

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

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

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

Square \sqrt{2}. Square 4.

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

Subtract 16 from 2 to get -14.

$$\frac{\sqrt{3}\left(\sqrt{2}+4\right)}{-14}$$

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

$$\frac{\sqrt{3}\sqrt{2}+4\sqrt{3}}{-14}$$

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

$$\frac{\sqrt{6}+4\sqrt{3}}{-14}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{6}-4\sqrt{3}}{14}$$