Solve 1/2-sqrt{2}+1/sqrt{2-1}):1/2

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

Evaluate

$$\frac{3\sqrt{2}}{2}+2\approx 4.121320344$$

Short Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

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

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

Consider \left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{2}. Square 1.

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

Subtract 1 from 2 to get 1.

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

Anything divided by one gives itself.

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

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

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

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

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

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

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

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

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