Solve 1/2-sqrt{2}-1

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

Evaluate

$$\frac{\sqrt{2}}{2}\approx 0.707106781$$

Short Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

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

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

Since \frac{2+\sqrt{2}}{2} and \frac{2}{2} have the same denominator, subtract them by subtracting their numerators.

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

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

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