Solve frac{sqrt{2}}{4}*frac{sqrt{2}}{2}+(1/2)^2-1/2*sqrt{3}+2*frac{sqrt{2}}{2}-(frac{sqrt{3}}{3})

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

Evaluate

$$-\frac{\sqrt{3}}{2}+\sqrt{2}+\frac{1}{2}\approx 1.048188159$$

Short Solution Steps

Multiply \frac{\sqrt{2}}{4} times \frac{\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

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

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

Cancel out 2 and 2.

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

Combine -\frac{\sqrt{3}}{6} and -\frac{\sqrt{3}}{3} to get -\frac{1}{2}\sqrt{3}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4\times 2 and 4 is 2\times 4. Multiply \frac{1}{4} times \frac{2}{2}.

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

Since \frac{\sqrt{2}\sqrt{2}}{2\times 4} and \frac{2}{2\times 4} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \sqrt{2}\sqrt{2}+2.

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

Do the calculations in 2+2.

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

Multiply 2 and 4 to get 8.

$$\frac{4}{8}-\frac{1}{2}\sqrt{3}+\sqrt{2}$$

Reduce the fraction \frac{4}{8} to lowest terms by extracting and canceling out 4.

$$\frac{1}{2}-\frac{1}{2}\sqrt{3}+\sqrt{2}$$

Factor

$$\frac{2 \sqrt{2} + 1 - \sqrt{3}}{2} = 1.0481881585886565$$