Question

How do you find the square root of 361?

Answer 1

`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`