How do you work out square roots?

Answer 1

There is Newton's iterative method,
x1 = x0 + (Y - x0^2)/(2x0), or
x1 = (Y + x0^2)/(2x0),
Where Y is the value you want the square root of,
x0 is your 1st guess,
x1 = refined value.
Replace x0 with x1 and recalculate until you have the desired accuracy.

there is another method which resembles long division, which is based on (10a + b)^2 = 100a^2 + 2*10ab + b^2:
Suppose you happen to want the square root of 1393.5289 (which happens to have an exact root).
Set up the process as follows, separating the number into groups of 2 digits on either side of the decimal point:
√13 93.52 89
find the largest square smaller than the leftmost 1 or 2 digits, and write its root above them:
... 3
√13 93.52 89
Enter the square of this number beneath and subtract, bringing down the next pair of digits:
... 3
√13 93.52 89
... 9
.. ------
. | 4 93
Multiply your "answer by 2 and use this as the 1st digit of a "divisor" to estimate the next digit of the answer:
...... 3. 7
...√13 93.52 89
...... 9
..... ------
.6_| 4 93
Fill the digit into the _ and multiply by it. Your product must be less than the new dividend:
...... 3. 7
...√13 93.52 89
...... 9
..... ------
.67| 4 93
.. 7| 4 69
Subtract, bring doun the next 2 digits and calculate your new divisor (Remember to keep track of your decimal point):
...... 3. 7 .
...√13 93.52 89
...... 9
..... ------
.67| 4 93
.. 7| 4 69
..... .---------
74_|.. 24.52
Repeating the process:
...... 3. 7 , 3
...√13 93,52 89
...... 9
..... ------
.67| 4 93
.. 7| 4 69
.... . ---------
...743|24.52
...... 3|22.92

...... 3. 7 , 3
...√13 93,52 89
...... 9
..... ------
.67| 4 93
.. 7| 4 69
... . ----------
...743|24.52
...... 3|22.29
..... ... -----------
... 746_|2.23 89

...... 3. 7 , 3.. 3
...√13 93,52 89
...... 9
..... ------
.67| 4 93
.. 7| 4 69
...... ---------
...743|24.52
...... 3|22.29
..... ... -----------
... 7463|2.23 89
......... 3|2.23 89
..... ... . ---------

Answer 2

It's very complicated to describe here but in general terms, you separate the number into pairs from the right. You note the square root of the first pair, subtract that from the number and then copy down the next pair and repeat the operation. You also add the individual square roots - no. I'm sorry. It is too complicated for me to do now but I'll put this on my watchlist and when I have some more time, I'll write out an example for you, unless someone comes up with a better way. Good Luck.

Incidentally, the square root of 400 is 20, easy but what about 4000? That's where Someone's solution falls down

I see Helmut's got there before me but here are another couple of examples.

Take 770884. Pair off from the decimal point or from the right if a whole number. 77 08 84. Take the first pair 77 and find the highest sqare root to go in it. In this case 8 so 8 is the first digit of the sqare root. Subtract 64 from 77 thus:

77 08 84
-64
=13 {note here 8}
then bring down next pair thus

13 08 {add 8 to 8 giving 16}
Find a digit which when put on the end of 16 and then multiplied, gives a number that goes into 1308, in this case 7 (the second digit of the square root)

so 167 x 7 gives 1169 {here note 167}
so 1308
- 1169
= 139

Bring down next pair so
139 84 {add 7 to 167 = 174}

find a digit that put on the end of 174 and multiplies to give a number that goes into 13984, in this case 8, the third digit of the square root
so1748x8=13984
so 13984
- 13984
= 0 and so 878 is the square root of 770884


77088 separate into pairs 7 70 88

7-4=3 (2x2=4 so 2 is the first digit0
bring down next pair
3 70 {add2+2 and find a digit when put on the end of 4 gives you a number that goes into 370}

47X7= 329 {47+7=54}
370
- 329
=41

bring down next pair 88 so
{continue as above, 547x7= 3829 so 7 is third digit of the sqare root}
4188
-3829
= 359

bring down next pair which as we have come to the end of the number must be the first two decimals

359.00 {547=7=554}

a digit to go on the end of 554 which when multiplied etc, is .6

332.76, which taken from 359.00 is 26.24 so by now you can tell that the square root is not going to be a whole number and you can continue as above to the number of decimal places you require. by now we have 277.6 which squared gives 77061.76

Hope you can follow my working out. The figures and calculations I have out in curly brackets you would normally put on the right side of the page and I used those two examples to show that six and five digit numbers have vastly different square roots. You can see that when we had time, we didn't need calculators (and I think our brains benefit more from exercise that our fingers do)

.

Answer 3

memorize 1-12(1-144)(numbers and their square roots)

for square root of 400

it would be 20

you can figure that out because the square root of a 400 can also be writen as the square root of 100 times the square root of 4(100*4=400)

and if u memorized 1-12, u would know 10*10=100 ans that 2*2=4

Answer 4

Use a calculator or if you know how do square a number then it should be easy enough to do the square root.
The square root of:
1 = 1
4 = 2
9 = 3
16 = 4
25 = 5
36 = 6
49 = 7
64 = 8
81 = 9
100 = 10
121 = 11
144 = 12
169 = 13 etc

Answer 5

In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. Every non-negative real number x has a unique non-negative square root, called the principal square root and denoted . For example, the principal square root of 9 is 3 (denoted ) because . The other square root of 9 (not the principal square root) is −3.

Square roots often arise when solving quadratic equations, or equations of the form ax2 + bx + c = 0, due to the variable x being squared.

Per the fundamental theorem of algebra, there are two solutions to the equation defining the square roots of any number (although these roots may not be distinct, as in the square root of zero). For a positive real number, the two square roots are the principal square root and the negative square root (denoted ). Together, the principal and negative square roots of a number are denoted . For negative real numbers, the concept of imaginary and complex numbers has been developed to provide a mathematical framework to deal with the results. Square roots of objects other than numbers can also be defined.

Square roots of integers that are not perfect squares are always irrational numbers, i.e., numbers not expressible as a ratio of two integers. For example, cannot be written exactly as m/n, where n and m are integers. Nonetheless, it is exactly the length of the diagonal of a square with side length 1. This has been known since ancient times, with the discovery that is irrational attributed to Hippasus, a disciple of Pythagoras. (See square root of 2 for proofs)

The square root symbol () was first used during the 16th century. It has been suggested that it originated as an altered form of lowercase r, representing the Latin radix (meaning "root").

Answer 6

First list out all the factors.Eg square root of 100
2
2
5
5
For two 2 You take one.
For two 5 you take out one
2*5=10

Answer 7

Most important is to break your number under the square root into perfect squares and/or prime factors.

For example SQRT60 = SQRT(4*15)
SQRT 4 = 2 and 15 cannot be rooted with whole numbers, so answer is 2SQRT15

Ex: SQRT80= SQRT(2*2*2*2*5)= (SQRT2*2)(SQRT2*2)(SQRT5)
= 4SQRT5

Hope this helps. If not, see the following site:

http://www.algebra-online.com/simplifying-square-roots-1.htm

Answer 8

u seem to be a beginner...
now take a perfect square number, eg. 100
now break it into prime factors, i.e., in this case 2*2*5*5
=(2^2)*(5^2)
Here i am using ^ for "to the power"
Now divide each of the powers by 2.
in this case u will get 2*5=10
hence the sq. root of 100 is10.

Answer 9

you may want to simplify sq. root 20 into 2 sq. root of 5 and sq. root of 8 into 2 sq. root of two consequently, you may want to component out 2, which leaves 2(sq. root of 5 + sq. root of two)^2 by simplifying extra, 2(5+2square root of 5 * sq. root of two + 2) 2(7+ 2 sq. root of 10) consequently the merely good answer might want to be: 14+ 4square root of 10

Answer 10

don't know about square roots,
but the roots of next doors tree are causing concern