Solve 1/sqrt{2+sqrt{4}}

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

Evaluate

$$-\frac{\sqrt{2}}{2}+1\approx 0.292893219$$

Solution Steps

Calculate the square root of 4 and get 2.

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

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

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

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}.

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

Square \sqrt{2}. Square 2.

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

Subtract 4 from 2 to get -2.

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

Multiply both numerator and denominator by -1.

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