Solve sqrt{1+1/12+1/2^2}

$$\sqrt { 1 + \frac { 1 } { 12 } + \frac { 1 } { 2 ^ { 2 } } }$$

Evaluate

$$\frac{2\sqrt{3}}{3}\approx 1.154700538$$

Solution Steps

Convert 1 to fraction \frac{12}{12}.

$$\sqrt{\frac{12}{12}+\frac{1}{12}+\frac{1}{2^{2}}}$$

Since \frac{12}{12} and \frac{1}{12} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{12+1}{12}+\frac{1}{2^{2}}}$$

Add 12 and 1 to get 13.

$$\sqrt{\frac{13}{12}+\frac{1}{2^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{\frac{13}{12}+\frac{1}{4}}$$

Least common multiple of 12 and 4 is 12. Convert \frac{13}{12} and \frac{1}{4} to fractions with denominator 12.

$$\sqrt{\frac{13}{12}+\frac{3}{12}}$$

Since \frac{13}{12} and \frac{3}{12} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{13+3}{12}}$$

Add 13 and 3 to get 16.

$$\sqrt{\frac{16}{12}}$$

Reduce the fraction \frac{16}{12} to lowest terms by extracting and canceling out 4.

$$\sqrt{\frac{4}{3}}$$

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

$$\frac{\sqrt{4}}{\sqrt{3}}$$

Calculate the square root of 4 and get 2.

$$\frac{2}{\sqrt{3}}$$

Rationalize the denominator of \frac{2}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{3}}{3}$$