Solve sqrt{frac{left(11-53right)^2+left(12-53right)^2+left(13-53right)^2+left(14-53right)^2+{left(15-53right)}^2}{53}}

$$\sqrt{ \frac{ { \left(11-53 \right) }^{ 2 } + { \left(12-53 \right) }^{ 2 } + { \left(13-53 \right) }^{ 2 } + { \left(14-53 \right) }^{ 2 } + { \left(15-53 \right) }^{ 2 } }{ 53 } }$$

Evaluate

$$\frac{3\sqrt{47170}}{53}\approx 12.293578628$$

Solution Steps

Subtract 53 from 11 to get -42.

$$\sqrt{\frac{\left(-42\right)^{2}+\left(12-53\right)^{2}+\left(13-53\right)^{2}+\left(14-53\right)^{2}+\left(15-53\right)^{2}}{53}}$$

Calculate -42 to the power of 2 and get 1764.

$$\sqrt{\frac{1764+\left(12-53\right)^{2}+\left(13-53\right)^{2}+\left(14-53\right)^{2}+\left(15-53\right)^{2}}{53}}$$

Subtract 53 from 12 to get -41.

$$\sqrt{\frac{1764+\left(-41\right)^{2}+\left(13-53\right)^{2}+\left(14-53\right)^{2}+\left(15-53\right)^{2}}{53}}$$

Calculate -41 to the power of 2 and get 1681.

$$\sqrt{\frac{1764+1681+\left(13-53\right)^{2}+\left(14-53\right)^{2}+\left(15-53\right)^{2}}{53}}$$

Add 1764 and 1681 to get 3445.

$$\sqrt{\frac{3445+\left(13-53\right)^{2}+\left(14-53\right)^{2}+\left(15-53\right)^{2}}{53}}$$

Subtract 53 from 13 to get -40.

$$\sqrt{\frac{3445+\left(-40\right)^{2}+\left(14-53\right)^{2}+\left(15-53\right)^{2}}{53}}$$

Calculate -40 to the power of 2 and get 1600.

$$\sqrt{\frac{3445+1600+\left(14-53\right)^{2}+\left(15-53\right)^{2}}{53}}$$

Add 3445 and 1600 to get 5045.

$$\sqrt{\frac{5045+\left(14-53\right)^{2}+\left(15-53\right)^{2}}{53}}$$

Subtract 53 from 14 to get -39.

$$\sqrt{\frac{5045+\left(-39\right)^{2}+\left(15-53\right)^{2}}{53}}$$

Calculate -39 to the power of 2 and get 1521.

$$\sqrt{\frac{5045+1521+\left(15-53\right)^{2}}{53}}$$

Add 5045 and 1521 to get 6566.

$$\sqrt{\frac{6566+\left(15-53\right)^{2}}{53}}$$

Subtract 53 from 15 to get -38.

$$\sqrt{\frac{6566+\left(-38\right)^{2}}{53}}$$

Calculate -38 to the power of 2 and get 1444.

$$\sqrt{\frac{6566+1444}{53}}$$

Add 6566 and 1444 to get 8010.

$$\sqrt{\frac{8010}{53}}$$

Rewrite the square root of the division \sqrt{\frac{8010}{53}} as the division of square roots \frac{\sqrt{8010}}{\sqrt{53}}.

$$\frac{\sqrt{8010}}{\sqrt{53}}$$

Factor 8010=3^{2}\times 890. Rewrite the square root of the product \sqrt{3^{2}\times 890} as the product of square roots \sqrt{3^{2}}\sqrt{890}. Take the square root of 3^{2}.

$$\frac{3\sqrt{890}}{\sqrt{53}}$$

Rationalize the denominator of \frac{3\sqrt{890}}{\sqrt{53}} by multiplying numerator and denominator by \sqrt{53}.

$$\frac{3\sqrt{890}\sqrt{53}}{\left(\sqrt{53}\right)^{2}}$$

The square of \sqrt{53} is 53.

$$\frac{3\sqrt{890}\sqrt{53}}{53}$$

To multiply \sqrt{890} and \sqrt{53}, multiply the numbers under the square root.

$$\frac{3\sqrt{47170}}{53}$$