Solve 2/2+sqrt{2}=

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

Evaluate

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

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

Cancel out 2 and 2.

$$2-\sqrt{2}$$