Solve sqrt{{left(17/6right)}^2+{left(17/3right)}^2+{left(17/6right)}^2}

$$\sqrt{ { \left( \frac{ 17 }{ 6 } \right) }^{ 2 } + { \left( \frac{ 17 }{ 3 } \right) }^{ 2 } + { \left( \frac{ 17 }{ 6 } \right) }^{ 2 } }$$

Evaluate

$$\frac{17\sqrt{6}}{6}\approx 6.940220938$$

Solution Steps

Calculate \frac{17}{6} to the power of 2 and get \frac{289}{36}.

$$\sqrt{\frac{289}{36}+\left(\frac{17}{3}\right)^{2}+\left(\frac{17}{6}\right)^{2}}$$

Calculate \frac{17}{3} to the power of 2 and get \frac{289}{9}.

$$\sqrt{\frac{289}{36}+\frac{289}{9}+\left(\frac{17}{6}\right)^{2}}$$

Least common multiple of 36 and 9 is 36. Convert \frac{289}{36} and \frac{289}{9} to fractions with denominator 36.

$$\sqrt{\frac{289}{36}+\frac{1156}{36}+\left(\frac{17}{6}\right)^{2}}$$

Since \frac{289}{36} and \frac{1156}{36} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{289+1156}{36}+\left(\frac{17}{6}\right)^{2}}$$

Add 289 and 1156 to get 1445.

$$\sqrt{\frac{1445}{36}+\left(\frac{17}{6}\right)^{2}}$$

Calculate \frac{17}{6} to the power of 2 and get \frac{289}{36}.

$$\sqrt{\frac{1445}{36}+\frac{289}{36}}$$

Since \frac{1445}{36} and \frac{289}{36} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{1445+289}{36}}$$

Add 1445 and 289 to get 1734.

$$\sqrt{\frac{1734}{36}}$$

Reduce the fraction \frac{1734}{36} to lowest terms by extracting and canceling out 6.

$$\sqrt{\frac{289}{6}}$$

Rewrite the square root of the division \sqrt{\frac{289}{6}} as the division of square roots \frac{\sqrt{289}}{\sqrt{6}}.

$$\frac{\sqrt{289}}{\sqrt{6}}$$

Calculate the square root of 289 and get 17.

$$\frac{17}{\sqrt{6}}$$

Rationalize the denominator of \frac{17}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{17\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{17\sqrt{6}}{6}$$