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

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

Evaluate

$$\frac{3\sqrt{2}}{4}-\frac{1}{2}\approx 0.560660172$$

Short Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

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

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

The square of \sqrt{2} is 2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

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

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

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

Cancel out 2 and 2.

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

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

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

Consider \left(2+\sqrt{2}\right)\left(2-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

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

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

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

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

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

$$\frac{2+\sqrt{2}}{4}-\frac{2\left(2-\sqrt{2}\right)}{4}$$

Since \frac{2+\sqrt{2}}{4} and \frac{2\left(2-\sqrt{2}\right)}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2+\sqrt{2}-2\left(2-\sqrt{2}\right)}{4}$$

Do the multiplications in 2+\sqrt{2}-2\left(2-\sqrt{2}\right).

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

Do the calculations in 2+\sqrt{2}-4+2\sqrt{2}.

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

Factor

$$\frac{3 \sqrt{2} - 2}{4} = 0.5606601717798214$$