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

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

Evaluate

$$\frac{\sqrt{6}}{2}\approx 1.224744871$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Multiply \frac{\sqrt{2}}{2} times \frac{\sqrt{3}}{2} by multiplying numerator times numerator and denominator times denominator.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply \frac{\sqrt{2}}{2} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply \frac{\sqrt{2}}{2} times \frac{\sqrt{3}}{2} by multiplying numerator times numerator and denominator times denominator.

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

Combine \frac{\sqrt{2}\sqrt{3}}{2\times 2} and \frac{\sqrt{2}\sqrt{3}}{2\times 2} to get 2\times \frac{\sqrt{2}\sqrt{3}}{2\times 2}.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply \frac{\sqrt{2}}{2} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Multiply 2 and 2 to get 4.

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

Multiply 2 and 2 to get 4.

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

Combine \frac{\sqrt{2}}{4} and -\frac{\sqrt{2}}{4} to get 0.

$$2\times \frac{\sqrt{2}\sqrt{3}}{2\times 2}$$

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

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

Multiply 2 and 2 to get 4.

$$2\times \frac{\sqrt{6}}{4}$$

Cancel out 4, the greatest common factor in 2 and 4.

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