Subjective Type

How to find the square root of 361?

Solution

$$361 = 19^{2}$$, 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 : $$361 \div 7 = 51$$ 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 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 $$19 \cdot 19 = 361$$

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