Question

25 upon root 40 minus root 80 root .5equal to 2.236 root 10 equal to 3.162

Answer 1

Hlw mate


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 √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=62 and √10≈3.162, so:

√361≈√360=√36⋅√10≈6⋅3.162≈19

Try 19⋅19=361 Yes

Memorise
Hey! I know that already: 361=192

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×b=(a+b2)2−(a−b2)2

For example:

23⋅27=252−22=625−4=621


Hope it helpful