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Question 631354: HELLO,
If anyone answer my questions i will be truly grateful
Find the square root of: 640.09
Find the square root of: 9682.56
Found 2 solutions by KMST, solver91311:
Answer by KMST(5396)   (Show Source):
You can put this solution on YOUR website! I assume that a pencil and paper calculation is required, because a calculator would give you the answer easily.
If the square roots are going to be rational numbers, the way to go is:
 and
If the square roots are going to be rational numbers, 64009 and 968256 must be perfect squares.
From there, I would try to factor the numbers in the square roots.
A complete prime factorization is not needed, but the idea is similar.
64009 is not divisible by 2 (it is not even) or 3 (digits do not add to a multiple of 3).
It is not divisible by 5 (does not end in 0 or 5) or by 7 (I tried dividing).
It is divisible by 11, because the sums of alternate digits differ by 11:
(6+0+9)-(4+0)=15-4=11
If is is a perfect square,  will be a factor, so I can divide by 11 twice.
I did, and I got 64009/11=5819 and 5819/11=529.
The number 529 sounded familiar, as if I had known it as a perfect square.
Since 529 ends in 9, it must be the square of a number that ends in 3 or 7.
That number be smaller than 25, because  ,
but larger than 20 because 
I tried 23 and found that 
So  and 
So 
968256 is divisible by 2, and if it is a perfect square it must be divisible by  .
So I divided by 4 once to get 968256/4=242064, and again to get 242064/4=60516,
and a third time to get 60516/4=15129, that is not divisible by 2 any more.
15129 is divisible by  and I divided 15129/9=1681.
If 1681 going to be a perfect square, it must be the square of a number that ends in 1 or 9, and is a bit larger than 40, because  .
I tried 41 and found that  so
41^2*9*4*4*4=41^2*3^2*64=41^2*3^2*8^2=(41*3*8)^2=984^2=968256 and 
So 
There is a general way to find square roots, that even allows you to get approximate values for irrational squre roots. It can be worked into a procedure sort of like long division, which I was taught in school many, many years ago. I hope that is not what was expected from you in the era of smartphones and tablet computers.
Answer by solver91311(24713)  (Show Source):
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