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

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

Evaluate

$$\sqrt{2}\approx 1.414213562$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

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

Calculate the square root of 1 and get 1.

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

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

$$\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\frac{1}{4}+\sqrt{\frac{1}{2}}-\frac{1}{4}$$

The square of \sqrt{2} is 2.

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

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

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

Calculate the square root of 1 and get 1.

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

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

$$\frac{\sqrt{2}}{2}+\frac{1}{4}+\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{1}{4}$$

The square of \sqrt{2} is 2.

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

Combine \frac{\sqrt{2}}{2} and \frac{\sqrt{2}}{2} to get \sqrt{2}.

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

Subtract \frac{1}{4} from \frac{1}{4} to get 0.

$$\sqrt{2}$$