Solve [1+1/sqrt{2}+1/2][1-1/sqrt{2}+1/2]

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

Evaluate

$$\frac{7}{4}=1.75$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

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

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

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

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

Add 2 and 1 to get 3.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

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

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

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

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

Add 2 and 1 to get 3.

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

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

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

Multiply \frac{3+\sqrt{2}}{2} and \frac{3+\sqrt{2}}{2} to get \left(\frac{3+\sqrt{2}}{2}\right)^{2}.

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

To raise \frac{3+\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\sqrt{2}\right)^{2}.

$$\frac{9+6\sqrt{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{9+6\sqrt{2}+2}{2^{2}}$$

Add 9 and 2 to get 11.

$$\frac{11+6\sqrt{2}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{11+6\sqrt{2}}{4}$$

Factor

$$\frac{7}{2 ^ {2}} = 1\frac{3}{4} = 1.75$$