Solve 3/-frac{1{sqrt{3}}-1/2sqrt{3}}

$$\frac{ 3 }{ - \frac{ 1 }{ \sqrt{ 3 } } - \frac{ 1 }{ 2 \sqrt{ 3 } } }$$

Evaluate

$$-2\sqrt{3}\approx -3.464101615$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{3}{-\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\frac{1}{2\sqrt{3}}}$$

The square of \sqrt{3} is 3.

$$\frac{3}{-\frac{\sqrt{3}}{3}-\frac{1}{2\sqrt{3}}}$$

Rationalize the denominator of \frac{1}{2\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

Combine -\frac{\sqrt{3}}{3} and -\frac{\sqrt{3}}{6} to get -\frac{1}{2}\sqrt{3}.

$$\frac{3}{-\frac{1}{2}\sqrt{3}}$$

Rationalize the denominator of \frac{3}{-\frac{1}{2}\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{3\sqrt{3}}{-\frac{1}{2}\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3\sqrt{3}}{-\frac{1}{2}\times 3}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\sqrt{3}}{-\frac{1}{2}}$$

Divide \sqrt{3} by -\frac{1}{2} by multiplying \sqrt{3} by the reciprocal of -\frac{1}{2}.

$$\frac{\sqrt{3}\times 2}{-1}$$

Anything divided by -1 gives its opposite.

$$-\sqrt{3}\times 2$$

Multiply -1 and 2 to get -2.

$$-2\sqrt{3}$$