Solve sqrt{(11/3)^2+(1/6)^2+(35/6)^2}

$$\sqrt { ( \frac { 11 } { 3 } ) ^ { 2 } + ( \frac { 1 } { 6 } ) ^ { 2 } + ( \frac { 35 } { 6 } ) ^ { 2 } }$$

Evaluate

$$\frac{\sqrt{190}}{2}\approx 6.892024376$$

Solution Steps

Calculate \frac{11}{3} to the power of 2 and get \frac{121}{9}.

$$\sqrt{\frac{121}{9}+\left(\frac{1}{6}\right)^{2}+\left(\frac{35}{6}\right)^{2}}$$

Calculate \frac{1}{6} to the power of 2 and get \frac{1}{36}.

$$\sqrt{\frac{121}{9}+\frac{1}{36}+\left(\frac{35}{6}\right)^{2}}$$

Least common multiple of 9 and 36 is 36. Convert \frac{121}{9} and \frac{1}{36} to fractions with denominator 36.

$$\sqrt{\frac{484}{36}+\frac{1}{36}+\left(\frac{35}{6}\right)^{2}}$$

Since \frac{484}{36} and \frac{1}{36} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{484+1}{36}+\left(\frac{35}{6}\right)^{2}}$$

Add 484 and 1 to get 485.

$$\sqrt{\frac{485}{36}+\left(\frac{35}{6}\right)^{2}}$$

Calculate \frac{35}{6} to the power of 2 and get \frac{1225}{36}.

$$\sqrt{\frac{485}{36}+\frac{1225}{36}}$$

Since \frac{485}{36} and \frac{1225}{36} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{485+1225}{36}}$$

Add 485 and 1225 to get 1710.

$$\sqrt{\frac{1710}{36}}$$

Reduce the fraction \frac{1710}{36} to lowest terms by extracting and canceling out 18.

$$\sqrt{\frac{95}{2}}$$

Rewrite the square root of the division \sqrt{\frac{95}{2}} as the division of square roots \frac{\sqrt{95}}{\sqrt{2}}.

$$\frac{\sqrt{95}}{\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{95}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{95}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{95}\sqrt{2}}{2}$$

To multiply \sqrt{95} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{190}}{2}$$