what is the method for finding out the square root of a number?

Question

a method that can work for all numbers.

Answer 1

let say take a square of 5

2x2 = 4
3x3 = 9

so it has to be between 2 and 3

2.5 x 2.5 = 6.25
still more than 5

2.4 x 2.4=5.76
still more 5

2.2 x 2.2=4.84
that's less than 5

2.25 x 2.25 = 5.0625
that's more than 5

2.24x2.24=5.0176

well, you get the ideal, that's how we find a square root of a number

Answer 2

To find out any square root of a number go this method:-
To find either the Least Common Multiple (LCM) or Greatest Common Factor (GCF) of two numbers, you always start out the same way: you find the prime factorizations of the two numbers. Then you put the factors into a nice neat grid of rows and columns, and then compare and contrast and take what you need. Here's how it works:
Find the GCF and LCM of 2940 and 3150.
First, you factor:





2940 = 2 × 2 × 3 × 5 × 7 × 7

3150 = 2 × 3 × 3 × 5 × 5 × 7

Now write these factors out all nice and neat, with the factors lined up according to occurrance:



Note how the factors are listed very orderly. This orderly listing will do most of the work for you!

The GCF is the biggest number that will divide into both 2940 and 3150. In other words, it's the number that contains all the common factors. So the GCF is the product of any and all factors that 2940 and 3150 share. Looking at the nice neat listing, you can see that the numbers both have a factor of 2; 2940 has a second copy, but 3150 does not, so you can only count the one copy toward your GCF. The numbers also share one copy of 3, one copy of 5, and one copy of 7.



Then the GCF is 2 × 3 × 5 × 7 = 210.

On the other hand, the LCM is the smallest number that both 2940 and 3150 will divide into. That is, it is the smallest number that contains both 2940 and 3150, that both numbers fit in to. Then it will be the smallest number that contains one of every factor in these two numbers. Looking back at the listing, you'll see that 3150 has one copy of the factor of 2; 2940 has two copies. Since the LCM must contain all factors of each number, the LCM must contain both copies of 2. However, to avoid overduplication, the LCM does not need three copies, because neither 2940 nor 3150 contains three copies.

This fact often causes confusion, so let's spend a little extra time on this. Consider two smaller numbers, 4 and 8, and their LCM. The number 4 factors as 2 × 2; 8 factors as 2 × 2 × 2. The LCM needs only have three copies of 2, in order to be divisible by both 4 and 8. That is, the LCM is 8. You do not need to take the three copies of 2 from the 8, and then throw in two extra copies from the 4. This would give you 32. While 32 is a common multiple, because 4 and 8 both divide evenly into 32, 32 is not the LEAST (smallest) common multiple, because you over-duplicated the 2s when you threw in the extra copies from the 4. Again, let the nice neat listing keep track of things when the numbers get big.

So, the LCM of 2940 and 3150 must contain both copies of the factor 2. By the same reasoning, the LCM must contain both copies of 3, both copies of 5, and both copies of 7:



Then the LCM is 2 × 2 × 3 × 3 × 5 × 5 × 7 × 7 = 44,100.

Using this "factor" method of listing the prime factors neatly in a table, you can always easily find the LCM and GCF. Completely factor the numbers you are given, list the factors neatly, with only one factor for each column (you can have a 2s column, a 3s column, etc, but a 3 would never go in a 2s column), and then carry the needed factors down to the bottom row. For the GCF, you carry down only those factors that all the listings share; for the LCM, you carry down all the factors, regardless of how few numbers contained that factor in their listings.


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Find the LCM and GCF of 27, 90, and 84.
First, find the prime factorizations:



Then list things out neatly:



Then the GCF and the LCM are given by:



Then the GCF is 3 and the LCM is 3,780.

Find the GCF and LCM of 3, 6, and 8.
First factor and list:



Then the GCF and LCM are given by:



Note that 3, 6, and 8 share no common factors. While 3 and 6 share a factor, and 6 and 8 share a factor, there is no prime factor that all three of them share. Since 1 divides into everything, then the greatest common factor in this case is just 1. When 1 is the GCF, the numbers are said to be "relatively" prime; that is, they are prime, relative to each other.

Then the GCF is 1 and the LCM is 2 × 2 × 2 × 3 = 24.


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The GCF doesn't come up that much at this stage in mathematics, though some books use it for factoring polynomial expressions by having the student find the GCF of all the terms in the polynomial and divide this value out of every term. But the LCM comes up every time you need to find a common denominator for fractions. The factor technique I demonstrated above works even for polynomial fractions. (The other method for finding the LCM, the "listing" method, will not work for polynomials, which is why you need to learn the factor method.) If you need to find the LCM of two (or more) polynomials, you can do the exact same procedure as above:

Find the LCM of x3 + 5x2 + 6x and 2x3 + 4x2.
First I factor the polynomials: x3 + 5x2 + 6x = x(x2 + 5x + 6) = x(x + 2)(x + 3), and 2x3 + 4x2 = 2x2(x + 2). Then I list these factors out, nice and neat:



Then the LCM is the product of one entry from each column:



I take two copies of "x", because 2x3 + 4x2 contains two copies. I don't need three copies of "x", because neither polynomial contains three copies. I need only one copy of x + 2, because neither polynomial contains more than just the one copy. I need the 2 for the second polynomial and the x + 3 for the first polynomial.

The LCM is 2x2(x + 2)(x + 3).

By the way, while you always multiply the factors together when you're finding the common denominators for regular fractions, you almost always want to leave the common denominators for polynomial fractions in factored form. That is, you'll need to multiply and simplify across the top, but don't multiply anything together across the bottom (in this case, don't multiply the x + 2 and the x + 3 to get x2 + 5x + 6). Remember: you'll still need to try to reduce the polynomial fraction when you're done simplifying across the top, so you'll need the bottom in factored form in the end, anyway.

Answer 3

For a quick estimate (like if i dont have a calculator and need it off the top of my head)...
I find the 2 nearest squares to the number, the one before and the one after....
and i take the ratio of the distance between the two that one is for the decimal...

ie:
what is the square root of 922?
well the two nearest squares are 30 and 31.
(900 and 961)

so i know that the root of 922 is at least 30
the decimal is 22/61 (approximately)

so the root of 922 is 30 + (22/61) = 30.36

but in reality 30.36 squared is 921.726


to find the actual square root.. i dont know

Answer 4

Decades ago, they used to actually teach a method for finding the square root by hand that was roughly analogous to long division.

No point in it now that we have calculators. It was tedious as all hell. And that's coming from somebody who LIKES doing long division by hand.

Answer 5

if you are taking or have taken college level courses, there are some numerical analysis techniques that you can use to solve the square root of a number.

2 techniques that come to mind are the bisection method and/or the newton-raphson methods.

Answer 6

i'm not sure what exaclty you mean, but i think of smaller numbers like this, going one by one, usually if i end up with a really big number and i want to find the square root of it i use a calculator

2x2=4
3x3=9

Answer 7

I assume you mean find the square root by hand. Here is a link.

http://www.nist.gov/dads/HTML/squareRoot.html

Answer 8

Hi:
You can find a method in the next link:

http://www.nist.gov/dads/HTML/squareRoot.html

Answer 9

better guess = (1/2)(guess + number/guess)

it is iterative.

look up Babylonian Method

Answer 10

You can find all the numbers that can divided evenly in a number you need to divid it to.