Solve sqrt{2}+1-frac{1+sqrt{2}}{2sqrt{2}}=

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

Evaluate

$$\frac{3\sqrt{2}}{4}+\frac{1}{2}\approx 1.560660172$$

Short Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{2}+1 times \frac{4}{4}.

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

Since \frac{4\left(\sqrt{2}+1\right)}{4} and \frac{\left(1+\sqrt{2}\right)\sqrt{2}}{4} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 4\left(\sqrt{2}+1\right)-\left(1+\sqrt{2}\right)\sqrt{2}.

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

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

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

Factor

$$\frac{\sqrt{2} {(\sqrt{2} + 3)}}{4} = 1.5606601717798214$$