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

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

Evaluate

$$\frac{-4\sqrt{3}-3}{13}\approx -0.763707941$$

Solution Steps

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

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

Consider \left(\sqrt{3}-4\right)\left(\sqrt{3}+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}.

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

Square \sqrt{3}. Square 4.

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

Subtract 16 from 3 to get -13.

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

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

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

The square of \sqrt{3} is 3.

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