Solve 1+1/sqrt{2}+1/2*1-1/sqrt{2}+1/2

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

Evaluate

$$2$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Multiply \frac{1}{2} and 1 to get \frac{1}{2}.

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

Convert 1 to fraction \frac{2}{2}.

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

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

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

Add 2 and 1 to get 3.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Since \frac{3}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

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

Add 3 and 1 to get 4.

$$\frac{4}{2}$$

Divide 4 by 2 to get 2.

$$2$$

Factor

$$2$$