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

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

Evaluate

$$\sqrt{6}+3\approx 5.449489743$$

Solution Steps

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

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

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}.

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

Square \sqrt{6}. Square 2.

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

Subtract 4 from 6 to get 2.

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

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

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

The square of \sqrt{6} is 6.

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

Divide each term of 6+2\sqrt{6} by 2 to get 3+\sqrt{6}.

$$3+\sqrt{6}$$