How to extract square roots by algebraic Manipulation?

Question

How to extract square roots by algebraic Manipulation?

Answer 1

Only for a perfect square can you do so I think.  EIther that it would be very very very hard.

Say you had 625

you would just factor it out

so 625 you can see its 5

5 x 125

5 x 5 x 25

5x5 x 5 x5

5^4 = 25^2

Only way i know

Answer 2

There are at least 2 methods:

The first is a method which is similar to long division. It’s not that hard and can be found in (mostly older) algebra books. (It's not easy to describe it here). Now that calculators are available, this method isn’t used too much.

There is however, another method which is very nice, easy, and less well known. It is by using the following iterative process:

Xn+1 = (1/2) * ( Xn +A/ Xn) Where A is the number you want the square root of..

and Xo is an initial guess.

If you’re not familiar with this notation, it means the following:

You make an initial guess at the answer, call it Xo. Then plug this in as Xn and calculate Xn+1 .(which will be called X1) Then put this answer into the equation as the new Xn. and calculate the new Xn+1 (which will be called X2). Then repeat this process . Each new result will be closer to the sqrt(A). It can be shown (with some work) that this sequence does converge. If you take the limit as n goes to infinity, you get

X=(1/2)*( X + A/X ) which simplifies to x^2 = A.

Here is an example: Find sqrt(2). So A=2. Guess the answer is = 1 ( = Xo ). Plug in and find that X1 = 1.5. Now take this result and plug in to find X2 = 1.4166. This is already correct to 2 decimal places. Now plug in 1.4166 and find X3 = 1.41421 which is correct to 5 decimal places. So in only 3 steps we have an excellent approximation to sqrt(2).

(This is from C.R.Buck, Advanced Calculus, 3rd Ed. pg 50.)

Answer 3

The square root of a number can be found by two methods.One is by factorisation,start with the smallest number two,divide by it and if the number is even remainder will be zero and if odd than ignore two and proceed to divide by three. Repeat the division till the number is divided into small pieces. say number is 196, dividing by 2 gives98 so write it as 2*98, again divide by 2 and now it becomes 2*2*49

49 cant be divided by 2,3,5 and is divisible by 7 so number is written as 2*2*7*7mmake pairs and there are two pairs of 2 and 7 so square root is 2*7=14

second method is division method196 is to be paird from the end side,so 96 f0rm a pair and 1 is left alone,1 can be divided by 1 only so 1 is quotent and remainder zero,take now the pair which is 96,divided it by taking double of 1 that is 2 and another number out of 0 to 9 so that 96 is divisble and no remainder is ther such number is 4, so divisor is 24 and quotent is4, sothat 24*4=96 and qoutient is 14(1 earlier and 4 from this step. Thus square root is 14