Solve frac{(sqrt{6}-sqrt{3})(sqrt{2}+1)}{sqrt{12}}-(-frac{sqrt{2}}{2})

$$\frac { ( \sqrt { 6 } - \sqrt { 3 } ) ( \sqrt { 2 } + 1 ) } { \sqrt { 12 } } - ( - \frac { \sqrt { 2 } } { 2 } )$$

Evaluate

$$\frac{\sqrt{2}+1}{2}\approx 1.207106781$$

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}.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

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

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

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}.

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

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

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

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

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

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

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

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

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

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

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

Reduce the fraction \frac{3}{6} to lowest terms by extracting and canceling out 3.

$$\frac{1}{2}-\left(-\frac{\sqrt{2}}{2}\right)$$

Multiply -1 and -1 to get 1.

$$\frac{1}{2}+\frac{\sqrt{2}}{2}$$

Since \frac{1}{2} and \frac{\sqrt{2}}{2} have the same denominator, add them by adding their numerators.

$$\frac{1+\sqrt{2}}{2}$$

Factor

$$\frac{\sqrt{2} + 1}{2} = 1.2071067811865475$$