Solve sqrt{(1/2)^2+3/10}

$$\sqrt { ( \frac { 1 } { 2 } ) ^ { 2 } + \frac { 3 } { 10 } }$$

Evaluate

$$\frac{\sqrt{55}}{10}\approx 0.741619849$$

Solution Steps

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\sqrt{\frac{1}{4}+\frac{3}{10}}$$

Least common multiple of 4 and 10 is 20. Convert \frac{1}{4} and \frac{3}{10} to fractions with denominator 20.

$$\sqrt{\frac{5}{20}+\frac{6}{20}}$$

Since \frac{5}{20} and \frac{6}{20} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{5+6}{20}}$$

Add 5 and 6 to get 11.

$$\sqrt{\frac{11}{20}}$$

Rewrite the square root of the division \sqrt{\frac{11}{20}} as the division of square roots \frac{\sqrt{11}}{\sqrt{20}}.

$$\frac{\sqrt{11}}{\sqrt{20}}$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{\sqrt{11}}{2\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{11}}{2\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{11}\sqrt{5}}{2\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{11}\sqrt{5}}{2\times 5}$$

To multiply \sqrt{11} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{55}}{2\times 5}$$

Multiply 2 and 5 to get 10.

$$\frac{\sqrt{55}}{10}$$