Solve =frac{sqrt{3}+sqrt{2}+1}{2(2+sqrt{6})}*frac{2-sqrt{6}}{2-sqrt{6}}

$$= \frac { \sqrt { 3 } + \sqrt { 2 } + 1 } { 2 ( 2 + \sqrt { 6 } ) } \times \frac { 2 - \sqrt { 6 } } { 2 - \sqrt { 6 } }$$

Evaluate

$$\frac{\sqrt{2}+\sqrt{6}-2}{4}\approx 0.465925826$$

Solution Steps

Divide 2-\sqrt{6} by 2-\sqrt{6} to get 1.

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

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

$$\frac{\sqrt{3}+\sqrt{2}+1}{4+2\sqrt{6}}\times 1$$

Rationalize the denominator of \frac{\sqrt{3}+\sqrt{2}+1}{4+2\sqrt{6}} by multiplying numerator and denominator by 4-2\sqrt{6}.

$$\frac{\left(\sqrt{3}+\sqrt{2}+1\right)\left(4-2\sqrt{6}\right)}{\left(4+2\sqrt{6}\right)\left(4-2\sqrt{6}\right)}\times 1$$

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

$$\frac{\left(\sqrt{3}+\sqrt{2}+1\right)\left(4-2\sqrt{6}\right)}{4^{2}-\left(2\sqrt{6}\right)^{2}}\times 1$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(\sqrt{3}+\sqrt{2}+1\right)\left(4-2\sqrt{6}\right)}{16-\left(2\sqrt{6}\right)^{2}}\times 1$$

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

$$\frac{\left(\sqrt{3}+\sqrt{2}+1\right)\left(4-2\sqrt{6}\right)}{16-2^{2}\left(\sqrt{6}\right)^{2}}\times 1$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(\sqrt{3}+\sqrt{2}+1\right)\left(4-2\sqrt{6}\right)}{16-4\left(\sqrt{6}\right)^{2}}\times 1$$

The square of \sqrt{6} is 6.

$$\frac{\left(\sqrt{3}+\sqrt{2}+1\right)\left(4-2\sqrt{6}\right)}{16-4\times 6}\times 1$$

Multiply 4 and 6 to get 24.

$$\frac{\left(\sqrt{3}+\sqrt{2}+1\right)\left(4-2\sqrt{6}\right)}{16-24}\times 1$$

Subtract 24 from 16 to get -8.

$$\frac{\left(\sqrt{3}+\sqrt{2}+1\right)\left(4-2\sqrt{6}\right)}{-8}\times 1$$

Express \frac{\left(\sqrt{3}+\sqrt{2}+1\right)\left(4-2\sqrt{6}\right)}{-8}\times 1 as a single fraction.

$$\frac{\left(\sqrt{3}+\sqrt{2}+1\right)\left(4-2\sqrt{6}\right)}{-8}$$

Apply the distributive property by multiplying each term of \sqrt{3}+\sqrt{2}+1 by each term of 4-2\sqrt{6}.

$$\frac{4\sqrt{3}-2\sqrt{3}\sqrt{6}+4\sqrt{2}-2\sqrt{2}\sqrt{6}+4-2\sqrt{6}}{-8}$$

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}.

$$\frac{4\sqrt{3}-2\sqrt{3}\sqrt{3}\sqrt{2}+4\sqrt{2}-2\sqrt{2}\sqrt{6}+4-2\sqrt{6}}{-8}$$

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

$$\frac{4\sqrt{3}-2\times 3\sqrt{2}+4\sqrt{2}-2\sqrt{2}\sqrt{6}+4-2\sqrt{6}}{-8}$$

Multiply -2 and 3 to get -6.

$$\frac{4\sqrt{3}-6\sqrt{2}+4\sqrt{2}-2\sqrt{2}\sqrt{6}+4-2\sqrt{6}}{-8}$$

Combine -6\sqrt{2} and 4\sqrt{2} to get -2\sqrt{2}.

$$\frac{4\sqrt{3}-2\sqrt{2}-2\sqrt{2}\sqrt{6}+4-2\sqrt{6}}{-8}$$

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}.

$$\frac{4\sqrt{3}-2\sqrt{2}-2\sqrt{2}\sqrt{2}\sqrt{3}+4-2\sqrt{6}}{-8}$$

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

$$\frac{4\sqrt{3}-2\sqrt{2}-2\times 2\sqrt{3}+4-2\sqrt{6}}{-8}$$

Multiply -2 and 2 to get -4.

$$\frac{4\sqrt{3}-2\sqrt{2}-4\sqrt{3}+4-2\sqrt{6}}{-8}$$

Combine 4\sqrt{3} and -4\sqrt{3} to get 0.

$$\frac{-2\sqrt{2}+4-2\sqrt{6}}{-8}$$