# How do you find the square root of 361?

Sep 8, 2015

$361 = {19}^{2}$, 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: $361 \div 7 = 51$ with remainder $4$.
$11$ : No: $361 \div 11 = 32$ with remainder $9$.
$13$ : No: $361 \div 13 = 27$ with remainder $10$.
$17$ : No: $361 \div 17 = 21$ 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 method.

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:

$\sqrt{361} \approx \sqrt{360} = \sqrt{36} \cdot \sqrt{10} \approx 6 \cdot 3.162 \approx 19$

Try $19 \cdot 19 = 361$ 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:

$a \times b = {\left(\frac{a + b}{2}\right)}^{2} - {\left(\frac{a - b}{2}\right)}^{2}$

For example:

$23 \cdot 27 = {25}^{2} - {2}^{2} = 625 - 4 = 621$