Solve frac{sqrt{6}}{6+2sqrt{6}}=

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

Evaluate

$$\frac{\sqrt{6}}{2}-1\approx 0.224744871$$

Solution Steps

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

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

Consider \left(6+2\sqrt{6}\right)\left(6-2\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate 6 to the power of 2 and get 36.

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

Expand \left(2\sqrt{6}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{6} is 6.

$$\frac{\sqrt{6}\left(6-2\sqrt{6}\right)}{36-4\times 6}$$

Multiply 4 and 6 to get 24.

$$\frac{\sqrt{6}\left(6-2\sqrt{6}\right)}{36-24}$$

Subtract 24 from 36 to get 12.

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

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

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

The square of \sqrt{6} is 6.

$$\frac{6\sqrt{6}-2\times 6}{12}$$

Multiply -2 and 6 to get -12.

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