What number does the square root of 82 fall between

Listing out a few elements of the set of perfect squares, we get the following values

{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...}

we can stop there because 82 is between 81 and 100

So if x^2 = 82, then x^2 is between 81 and 100 meaning

81 < x^2 < 100

apply the square root to all sides of the compound inequality and we get

9 < x < 10

So the square root of 81 is between 9 and 10

the value falls between 9 and 10