Solve sqrt{(2-4)^2+(3-4)^2+(4-4)^2+(7-4)^2/4}=

$$\sqrt { \frac { ( 2 - 4 ) ^ { 2 } + ( 3 - 4 ) ^ { 2 } + ( 4 - 4 ) ^ { 2 } + ( 7 - 4 ) ^ { 2 } } { 4 } } =$$

Evaluate

$$\frac{\sqrt{14}}{2}\approx 1.870828693$$

Solution Steps

Subtract 4 from 2 to get -2.

$$\sqrt{\frac{\left(-2\right)^{2}+\left(3-4\right)^{2}+\left(4-4\right)^{2}+\left(7-4\right)^{2}}{4}}$$

Calculate -2 to the power of 2 and get 4.

$$\sqrt{\frac{4+\left(3-4\right)^{2}+\left(4-4\right)^{2}+\left(7-4\right)^{2}}{4}}$$

Subtract 4 from 3 to get -1.

$$\sqrt{\frac{4+\left(-1\right)^{2}+\left(4-4\right)^{2}+\left(7-4\right)^{2}}{4}}$$

Calculate -1 to the power of 2 and get 1.

$$\sqrt{\frac{4+1+\left(4-4\right)^{2}+\left(7-4\right)^{2}}{4}}$$

Add 4 and 1 to get 5.

$$\sqrt{\frac{5+\left(4-4\right)^{2}+\left(7-4\right)^{2}}{4}}$$

Subtract 4 from 4 to get 0.

$$\sqrt{\frac{5+0^{2}+\left(7-4\right)^{2}}{4}}$$

Calculate 0 to the power of 2 and get 0.

$$\sqrt{\frac{5+0+\left(7-4\right)^{2}}{4}}$$

Add 5 and 0 to get 5.

$$\sqrt{\frac{5+\left(7-4\right)^{2}}{4}}$$

Subtract 4 from 7 to get 3.

$$\sqrt{\frac{5+3^{2}}{4}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{\frac{5+9}{4}}$$

Add 5 and 9 to get 14.

$$\sqrt{\frac{14}{4}}$$

Reduce the fraction \frac{14}{4} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{7}{2}}$$

Rewrite the square root of the division \sqrt{\frac{7}{2}} as the division of square roots \frac{\sqrt{7}}{\sqrt{2}}.

$$\frac{\sqrt{7}}{\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{7}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{7}\sqrt{2}}{2}$$

To multiply \sqrt{7} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{14}}{2}$$