Solve sqrt{1/2}+2sqrt{1/8}

$$\sqrt{ \frac{ 1 }{ 2 } } +2 \sqrt{ \frac{ 1 }{ 8 } }$$

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}}.

$$\frac{\sqrt{1}}{\sqrt{2}}+2\sqrt{\frac{1}{8}}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{2}}+2\sqrt{\frac{1}{8}}$$

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

$$\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+2\sqrt{\frac{1}{8}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{2}+2\sqrt{\frac{1}{8}}$$

Rewrite the square root of the division \sqrt{\frac{1}{8}} as the division of square roots \frac{\sqrt{1}}{\sqrt{8}}.

$$\frac{\sqrt{2}}{2}+2\times \frac{\sqrt{1}}{\sqrt{8}}$$

Calculate the square root of 1 and get 1.

$$\frac{\sqrt{2}}{2}+2\times \frac{1}{\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

Cancel out 4, the greatest common factor in 2 and 4.

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

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

$$\sqrt{2}$$