Question

How to find the square root of 361?

Answer 1

$361={19}^2$, so $\sqrt{361}=19$
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 pf 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 : $361=19\cdot 19$
So $\sqrt{361}=19$
Approximation by integers
$20\cdot 20=400$, 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 methos.
Try $19\cdot 19=361$ Yes.
Hmmm, I know some square roots already
I know $36=6^2$ and $\sqrt{10}\approx 3.162$, so:
$\sqrt{361}\approx \sqrt{360}=\sqrt{36}\cdot \sqrt{10}\approx 6.3162\approx 19$
Try $19\cdot 19=361$