Solve sqrt{frac{left(3.96-3.998right)^2+left(3.98-3.998right)^2+left(3.95-3.998right)^2+left(4.03-3.998right)^2+{left(3.99-3.998right)}^{2}+{left(4.03-3.998right)}^{2}+{left(4.02-3.998right)}^2+{left(3.99-3.998right)}^2+{left(4.04-3.998right)}^2+{left(3.99+3.998right)}^2}{90}}

$$\sqrt{ \frac{ { \left(3.96-3.998 \right) }^{ 2 } + { \left(3.98-3.998 \right) }^{ 2 } + { \left(3.95-3.998 \right) }^{ 2 } + { \left(4.03-3.998 \right) }^{ 2 } + { \left(3.99-3.998 \right) }^{ 2 } + { \left(4.03-3.998 \right) }^{ 2 } + { \left(4.02-3.998 \right) }^{ 2 } + { \left(3.99-3.998 \right) }^{ 2 } + { \left(4.04-3.998 \right) }^{ 2 } + { \left(3.99+3.998 \right) }^{ 2 } }{ 90 } }$$

Evaluate

$$\frac{\sqrt{398854}}{750}\approx 0.842065186$$

Solution Steps

Subtract 3.998 from 3.96 to get -0.038.

$$\sqrt{\frac{\left(-0.038\right)^{2}+\left(3.98-3.998\right)^{2}+\left(3.95-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Calculate -0.038 to the power of 2 and get 0.001444.

$$\sqrt{\frac{0.001444+\left(3.98-3.998\right)^{2}+\left(3.95-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Subtract 3.998 from 3.98 to get -0.018.

$$\sqrt{\frac{0.001444+\left(-0.018\right)^{2}+\left(3.95-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Calculate -0.018 to the power of 2 and get 0.000324.

$$\sqrt{\frac{0.001444+0.000324+\left(3.95-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Add 0.001444 and 0.000324 to get 0.001768.

$$\sqrt{\frac{0.001768+\left(3.95-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Subtract 3.998 from 3.95 to get -0.048.

$$\sqrt{\frac{0.001768+\left(-0.048\right)^{2}+\left(4.03-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Calculate -0.048 to the power of 2 and get 0.002304.

$$\sqrt{\frac{0.001768+0.002304+\left(4.03-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Add 0.001768 and 0.002304 to get 0.004072.

$$\sqrt{\frac{0.004072+\left(4.03-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Subtract 3.998 from 4.03 to get 0.032.

$$\sqrt{\frac{0.004072+0.032^{2}+\left(3.99-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Calculate 0.032 to the power of 2 and get 0.001024.

$$\sqrt{\frac{0.004072+0.001024+\left(3.99-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Add 0.004072 and 0.001024 to get 0.005096.

$$\sqrt{\frac{0.005096+\left(3.99-3.998\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Subtract 3.998 from 3.99 to get -0.008.

$$\sqrt{\frac{0.005096+\left(-0.008\right)^{2}+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Calculate -0.008 to the power of 2 and get 0.000064.

$$\sqrt{\frac{0.005096+0.000064+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Add 0.005096 and 0.000064 to get 0.00516.

$$\sqrt{\frac{0.00516+\left(4.03-3.998\right)^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Subtract 3.998 from 4.03 to get 0.032.

$$\sqrt{\frac{0.00516+0.032^{2}+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Calculate 0.032 to the power of 2 and get 0.001024.

$$\sqrt{\frac{0.00516+0.001024+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Add 0.00516 and 0.001024 to get 0.006184.

$$\sqrt{\frac{0.006184+\left(4.02-3.998\right)^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Subtract 3.998 from 4.02 to get 0.022.

$$\sqrt{\frac{0.006184+0.022^{2}+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Calculate 0.022 to the power of 2 and get 0.000484.

$$\sqrt{\frac{0.006184+0.000484+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Add 0.006184 and 0.000484 to get 0.006668.

$$\sqrt{\frac{0.006668+\left(3.99-3.998\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Subtract 3.998 from 3.99 to get -0.008.

$$\sqrt{\frac{0.006668+\left(-0.008\right)^{2}+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Calculate -0.008 to the power of 2 and get 0.000064.

$$\sqrt{\frac{0.006668+0.000064+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Add 0.006668 and 0.000064 to get 0.006732.

$$\sqrt{\frac{0.006732+\left(4.04-3.998\right)^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Subtract 3.998 from 4.04 to get 0.042.

$$\sqrt{\frac{0.006732+0.042^{2}+\left(3.99+3.998\right)^{2}}{90}}$$

Calculate 0.042 to the power of 2 and get 0.001764.

$$\sqrt{\frac{0.006732+0.001764+\left(3.99+3.998\right)^{2}}{90}}$$

Add 0.006732 and 0.001764 to get 0.008496.

$$\sqrt{\frac{0.008496+\left(3.99+3.998\right)^{2}}{90}}$$

Add 3.99 and 3.998 to get 7.988.

$$\sqrt{\frac{0.008496+7.988^{2}}{90}}$$

Calculate 7.988 to the power of 2 and get 63.808144.

$$\sqrt{\frac{0.008496+63.808144}{90}}$$

Add 0.008496 and 63.808144 to get 63.81664.

$$\sqrt{\frac{63.81664}{90}}$$

Expand \frac{63.81664}{90} by multiplying both numerator and the denominator by 100000.

$$\sqrt{\frac{6381664}{9000000}}$$

Reduce the fraction \frac{6381664}{9000000} to lowest terms by extracting and canceling out 32.

$$\sqrt{\frac{199427}{281250}}$$

Rewrite the square root of the division \sqrt{\frac{199427}{281250}} as the division of square roots \frac{\sqrt{199427}}{\sqrt{281250}}.

$$\frac{\sqrt{199427}}{\sqrt{281250}}$$

Factor 281250=375^{2}\times 2. Rewrite the square root of the product \sqrt{375^{2}\times 2} as the product of square roots \sqrt{375^{2}}\sqrt{2}. Take the square root of 375^{2}.

$$\frac{\sqrt{199427}}{375\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{199427}}{375\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{199427}\sqrt{2}}{375\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{199427}\sqrt{2}}{375\times 2}$$

To multiply \sqrt{199427} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{398854}}{375\times 2}$$

Multiply 375 and 2 to get 750.

$$\frac{\sqrt{398854}}{750}$$