Solve frac{6-sqrt{6}}{sqrt{12}-sqrt{2}}

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

Evaluate

$$\sqrt{3}\approx 1.732050808$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

The square of \sqrt{2} is 2.

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

Subtract 2 from 12 to get 10.

$$\frac{\left(6-\sqrt{6}\right)\left(2\sqrt{3}+\sqrt{2}\right)}{10}$$

Apply the distributive property by multiplying each term of 6-\sqrt{6} by each term of 2\sqrt{3}+\sqrt{2}.

$$\frac{12\sqrt{3}+6\sqrt{2}-2\sqrt{3}\sqrt{6}-\sqrt{6}\sqrt{2}}{10}$$

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}.

$$\frac{12\sqrt{3}+6\sqrt{2}-2\sqrt{3}\sqrt{3}\sqrt{2}-\sqrt{6}\sqrt{2}}{10}$$

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

$$\frac{12\sqrt{3}+6\sqrt{2}-2\times 3\sqrt{2}-\sqrt{6}\sqrt{2}}{10}$$

Multiply -2 and 3 to get -6.

$$\frac{12\sqrt{3}+6\sqrt{2}-6\sqrt{2}-\sqrt{6}\sqrt{2}}{10}$$

Combine 6\sqrt{2} and -6\sqrt{2} to get 0.

$$\frac{12\sqrt{3}-\sqrt{6}\sqrt{2}}{10}$$

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

$$\frac{12\sqrt{3}-\sqrt{2}\sqrt{3}\sqrt{2}}{10}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{12\sqrt{3}-2\sqrt{3}}{10}$$

Combine 12\sqrt{3} and -2\sqrt{3} to get 10\sqrt{3}.

$$\frac{10\sqrt{3}}{10}$$

Cancel out 10 and 10.

$$\sqrt{3}$$