How do you find the square root of 1849?

1 Answer
George C.
Jul 21, 2018

#sqrt(1849) = 43#

Explanation:

We could first seek to find the prime factorisation of #1849#, but as we shall see it is actually the square of a prime number, so that would be somewhat tedious.

Alternatively, let's split it into pairs of digits from the right to get:

#18"|"49#

Examining the leading #18#, note that it lies between #4^2# and #5^2#:

#4^2 = 16 < 18 < 25 = 5^2#

So:

#4 < sqrt(18) < 5#

and hence:

#40 < sqrt(1849) < 50#

To find a suitable correction, we can linearly interpolate between #40# and #50# to find:

#sqrt(1849) ~~ 40 + (50-40) * (1849 - 40^2)/(50^2-40^2)#

#color(white)(sqrt(1849)) ~~ 40 + 10 * (1849 - 1600)/(2500-1600)#

#color(white)(sqrt(1849)) ~~ 40 + 2490/900#

#color(white)(sqrt(1849)) ~~ 40 + 2.49 + 0.249 + 0.0249 +...#

#color(white)(sqrt(1849)) ~~ 42.76#

Hmmm... That's close to #43#, let's try #43^2#...

#43*43 = 40^2 + 2 * 40 * 3 + 3^2 = 1600+240+9 = 1849#

So:

#sqrt(1849) = 43#

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