##361 = 19^2## so ##sqrt(361) = 19##.
See explanation for a few methods…
Prime Factorisation
One of the best ways to attempt to find the square root of a whole number is to factor it into primes and identify pairs of identical factors. This is a bit tedious in the case of ##361## as we shall see:
Let’s try each prime in turn:
##2## : No: ##361## is not even.
##3## : No: The sum of the digits is not a multiple of ##3##.
##5## : No: The last digit of ##361## is not ##0## or ##5##.
##7## : No: ##361 -: 7 = 51## with remainder ##4##.
##11## : No: ##361 -: 11 = 32## with remainder ##9##.
##13## : No: ##361 -: 13 = 27## with remainder ##10##.
##17## : No: ##361 -: 17 = 21## with remainder ##4##.
##19## : Yes: ##361 = 19*19##
So ##sqrt(361) = 19##
Approximation by integers
##20*20 = 400## so that’s about ##10##% too large.
Subtract half that percentage from the approximation:
##20 – 5##% ##= 19##
The half that percentage bit is a form of Newton Raphson method.
Try ##19*19 = 361## Yes.
Hmmm I know some square roots already
I know ##36 = 6^2## and ##sqrt(10) ~~ 3.162## so:
##sqrt(361) ~~ sqrt(360) = sqrt(36) * sqrt(10) ~~ 6 * 3.162 ~~ 19##
Try ##19*19 = 361## Yes
Memorise
Hey! I know that already: ##361 = 19^2##
Knowing a few squares is useful for all sorts of mental calculation so I would recommend memorising them a bit. In fact you can multiply two odd or two even numbers using squares adding subtracting and halving as follows:
##a xx b = ((a+b)/2)^2 – ((a-b)/2)^2##
For example:
##23 * 27 = 25^2 – 2^2 = 625 – 4 = 621##