Question 139941
First, note that the numbers 25 and 36 are perfect squares. In other words, {{{5^2=25}}} and {{{6^2=36}}}. So since 31 is in between 25 and 36, this means that the square root of 31 is in between 5 and 6. In other words, since {{{25<31<36}}}, this means {{{5<sqrt(31)<6}}}. 



Notice how 31 is 6 units away from 25 and 5 units away from 36. So 31 is about the halfway point from 25 to 36. So a good approximation for {{{sqrt(31)}}} is 5.5 since 5.5 is halfway between 5 and 6.



So *[Tex \LARGE \sqrt{31}\approx 5.5]



If we take the square root of 31 with a calculator, we get {{{sqrt(31)=5.5677643628}}}. So this shows us that our approximation is very close.