Solve frac{sqrt{2}}{4+sqrt{2}}

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

Evaluate

$$\frac{2\sqrt{2}-1}{7}\approx 0.261203875$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{2}}{4+\sqrt{2}} by multiplying numerator and denominator by 4-\sqrt{2}.

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

Consider \left(4+\sqrt{2}\right)\left(4-\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}.

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

Square 4. Square \sqrt{2}.

$$\frac{\sqrt{2}\left(4-\sqrt{2}\right)}{16-2}$$

Subtract 2 from 16 to get 14.

$$\frac{\sqrt{2}\left(4-\sqrt{2}\right)}{14}$$

Use the distributive property to multiply \sqrt{2} by 4-\sqrt{2}.

$$\frac{4\sqrt{2}-\left(\sqrt{2}\right)^{2}}{14}$$

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{2}-2}{14}$$