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

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

Evaluate

$$2\left(\sqrt{6}+3\right)\approx 10.898979486$$

Solution Steps

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

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

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

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

Square \sqrt{6}. Square 2.

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

Subtract 4 from 6 to get 2.

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

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

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

The square of \sqrt{6} is 6.

$$\frac{2\times 6+4\sqrt{6}}{2}$$

Multiply 2 and 6 to get 12.

$$\frac{12+4\sqrt{6}}{2}$$