How do you find the square root of 361?

1 Answer
Sep 8, 2015

Answer:

#361 = 19^2#, so #sqrt(361) = 19#.

See explanation for a few methods...

Explanation:

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#