Solve frac{sqrt{3}}{6-4sqrt{3}}

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

Evaluate

$$-\frac{\sqrt{3}}{2}-1\approx -1.866025404$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}}{6-4\sqrt{3}} by multiplying numerator and denominator by 6+4\sqrt{3}.

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

Consider \left(6-4\sqrt{3}\right)\left(6+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}.

$$\frac{\sqrt{3}\left(6+4\sqrt{3}\right)}{6^{2}-\left(-4\sqrt{3}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{\sqrt{3}\left(6+4\sqrt{3}\right)}{36-\left(-4\sqrt{3}\right)^{2}}$$

Expand \left(-4\sqrt{3}\right)^{2}.

$$\frac{\sqrt{3}\left(6+4\sqrt{3}\right)}{36-\left(-4\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -4 to the power of 2 and get 16.

$$\frac{\sqrt{3}\left(6+4\sqrt{3}\right)}{36-16\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{3}\left(6+4\sqrt{3}\right)}{36-16\times 3}$$

Multiply 16 and 3 to get 48.

$$\frac{\sqrt{3}\left(6+4\sqrt{3}\right)}{36-48}$$

Subtract 48 from 36 to get -12.

$$\frac{\sqrt{3}\left(6+4\sqrt{3}\right)}{-12}$$

Use the distributive property to multiply \sqrt{3} by 6+4\sqrt{3}.

$$\frac{6\sqrt{3}+4\left(\sqrt{3}\right)^{2}}{-12}$$

The square of \sqrt{3} is 3.

$$\frac{6\sqrt{3}+4\times 3}{-12}$$

Multiply 4 and 3 to get 12.

$$\frac{6\sqrt{3}+12}{-12}$$