Solve sqrt{frac{left(100.5-500right)^2+left(300.5-500right)^2+left(500.5-500right)^2+left(700.5-500right)^2+{left(900.5-500right)}^2}{500}}

$$\sqrt{ \frac{ { \left(100.5-500 \right) }^{ 2 } + { \left(300.5-500 \right) }^{ 2 } + { \left(500.5-500 \right) }^{ 2 } + { \left(700.5-500 \right) }^{ 2 } + { \left(900.5-500 \right) }^{ 2 } }{ 500 } }$$

Evaluate

$$\frac{\sqrt{320001}}{20}\approx 28.284315442$$

Solution Steps

Subtract 500 from 100.5 to get -399.5.

$$\sqrt{\frac{\left(-399.5\right)^{2}+\left(300.5-500\right)^{2}+\left(500.5-500\right)^{2}+\left(700.5-500\right)^{2}+\left(900.5-500\right)^{2}}{500}}$$

Calculate -399.5 to the power of 2 and get 159600.25.

$$\sqrt{\frac{159600.25+\left(300.5-500\right)^{2}+\left(500.5-500\right)^{2}+\left(700.5-500\right)^{2}+\left(900.5-500\right)^{2}}{500}}$$

Subtract 500 from 300.5 to get -199.5.

$$\sqrt{\frac{159600.25+\left(-199.5\right)^{2}+\left(500.5-500\right)^{2}+\left(700.5-500\right)^{2}+\left(900.5-500\right)^{2}}{500}}$$

Calculate -199.5 to the power of 2 and get 39800.25.

$$\sqrt{\frac{159600.25+39800.25+\left(500.5-500\right)^{2}+\left(700.5-500\right)^{2}+\left(900.5-500\right)^{2}}{500}}$$

Add 159600.25 and 39800.25 to get 199400.5.

$$\sqrt{\frac{199400.5+\left(500.5-500\right)^{2}+\left(700.5-500\right)^{2}+\left(900.5-500\right)^{2}}{500}}$$

Subtract 500 from 500.5 to get 0.5.

$$\sqrt{\frac{199400.5+0.5^{2}+\left(700.5-500\right)^{2}+\left(900.5-500\right)^{2}}{500}}$$

Calculate 0.5 to the power of 2 and get 0.25.

$$\sqrt{\frac{199400.5+0.25+\left(700.5-500\right)^{2}+\left(900.5-500\right)^{2}}{500}}$$

Add 199400.5 and 0.25 to get 199400.75.

$$\sqrt{\frac{199400.75+\left(700.5-500\right)^{2}+\left(900.5-500\right)^{2}}{500}}$$

Subtract 500 from 700.5 to get 200.5.

$$\sqrt{\frac{199400.75+200.5^{2}+\left(900.5-500\right)^{2}}{500}}$$

Calculate 200.5 to the power of 2 and get 40200.25.

$$\sqrt{\frac{199400.75+40200.25+\left(900.5-500\right)^{2}}{500}}$$

Add 199400.75 and 40200.25 to get 239601.

$$\sqrt{\frac{239601+\left(900.5-500\right)^{2}}{500}}$$

Subtract 500 from 900.5 to get 400.5.

$$\sqrt{\frac{239601+400.5^{2}}{500}}$$

Calculate 400.5 to the power of 2 and get 160400.25.

$$\sqrt{\frac{239601+160400.25}{500}}$$

Add 239601 and 160400.25 to get 400001.25.

$$\sqrt{\frac{400001.25}{500}}$$

Expand \frac{400001.25}{500} by multiplying both numerator and the denominator by 100.

$$\sqrt{\frac{40000125}{50000}}$$

Reduce the fraction \frac{40000125}{50000} to lowest terms by extracting and canceling out 125.

$$\sqrt{\frac{320001}{400}}$$

Rewrite the square root of the division \sqrt{\frac{320001}{400}} as the division of square roots \frac{\sqrt{320001}}{\sqrt{400}}.

$$\frac{\sqrt{320001}}{\sqrt{400}}$$

Calculate the square root of 400 and get 20.

$$\frac{\sqrt{320001}}{20}$$