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

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

Evaluate

$$-4\sqrt{6}-6\sqrt{2}\approx -18.283240345$$

Solution Steps

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

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

Consider \left(\sqrt{3}-2\right)\left(\sqrt{3}+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{3}. Square 2.

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

Subtract 4 from 3 to get -1.

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

Anything divided by -1 gives its opposite.

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

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

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

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}.

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

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

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

Multiply 2 and 3 to get 6.

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

To find the opposite of 6\sqrt{2}+4\sqrt{6}, find the opposite of each term.

$$-6\sqrt{2}-4\sqrt{6}$$