Solve frac{2sqrt{3}}{sqrt{6}-2sqrt{3}}

$$\frac { 2 \sqrt { 3 } } { \sqrt { 6 } - 2 \sqrt { 3 } }$$

Evaluate

$$-\sqrt{2}-2\approx -3.414213562$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{3}}{\sqrt{6}-2\sqrt{3}} by multiplying numerator and denominator by \sqrt{6}+2\sqrt{3}.

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

Consider \left(\sqrt{6}-2\sqrt{3}\right)\left(\sqrt{6}+2\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}.

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

The square of \sqrt{6} is 6.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

Subtract 12 from 6 to get -6.

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

Divide 2\sqrt{3}\left(\sqrt{6}+2\sqrt{3}\right) by -6 to get -\frac{1}{3}\sqrt{3}\left(\sqrt{6}+2\sqrt{3}\right).

$$-\frac{1}{3}\sqrt{3}\left(\sqrt{6}+2\sqrt{3}\right)$$

Use the distributive property to multiply -\frac{1}{3}\sqrt{3} by \sqrt{6}+2\sqrt{3}.

$$-\frac{1}{3}\sqrt{3}\sqrt{6}-\frac{1}{3}\sqrt{3}\times 2\sqrt{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$-\frac{1}{3}\sqrt{3}\sqrt{6}-\frac{1}{3}\times 3\times 2$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

Cancel out 3 and 3.

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

Cancel out 3 and 3.

$$-\sqrt{2}-2$$