Solve 1/sqrt{2/2+1}

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

Evaluate

$$2-\sqrt{2}\approx 0.585786438$$

Solution Steps

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

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

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

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

Divide 1 by \frac{\sqrt{2}+2}{2} by multiplying 1 by the reciprocal of \frac{\sqrt{2}+2}{2}.

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

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

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

Consider \left(\sqrt{2}+2\right)\left(\sqrt{2}-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}.

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

Square \sqrt{2}. Square 2.

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

Subtract 4 from 2 to get -2.

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

Cancel out -2 and -2.

$$-\left(\sqrt{2}-2\right)$$

To find the opposite of \sqrt{2}-2, find the opposite of each term.

$$-\sqrt{2}-\left(-2\right)$$

The opposite of -2 is 2.

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

Factor

$$\sqrt{2} {(\sqrt{2} - 1)} = 0.585786438$$