Solve 4+sqrt{1/8}

$$4+ \sqrt{ \frac{ 1 }{ 8 } }$$

Evaluate

$$\frac{\sqrt{2}}{4}+4\approx 4.353553391$$

Short Solution Steps

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

$$4+\frac{\sqrt{1}}{\sqrt{8}}$$

Calculate the square root of 1 and get 1.

$$4+\frac{1}{\sqrt{8}}$$

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}.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{4}{4}.

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

Since \frac{4\times 4}{4} and \frac{\sqrt{2}}{4} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 4\times 4+\sqrt{2}.

$$\frac{16+\sqrt{2}}{4}$$