Can anyone tell me the easiest way to find a square root of a number?

Question

It seems like you have to know the number in your head in order to know a square root of a number. Is there a way to figure it out on paper? Like the square root of 289. I keep trying to figure it out on the calculator and having a problem. Thanks in advance.

Answer 1

Most calculators have a square root key. If they do not you can try the x^y key and put in 0.5 for y.

If you don't have either of these options then try the "Babylonian method" (you can google square root Babylonian Method or go to wiki link below) which converges very quickly to the real square root number.

Babylonian Method:
Try to find square root where N=265.

Our first guess could be 15 because we know that 15*15 = 225. But let's pretend we don't know that. Since there are 3 digits in this number, our first guess will be (num digits)^3 or 27.

x0 = 27
x1 = 1/2 * (x0 + N/x0) = 1/2 (27 + 265/27) = 18.41
x2 = 1/2 * (x1 + N/x1) = 1/2 (18.41 + 265/18.41) = 16.402
x3 = 1/2 * (x2 + N/x2) = 1/2 (16.402 + 265/16.402) = 16.2792
x4 = 1/2 * (x3 + N/x3) = 1/2 (16.2792 + 265/16.2792) = 16.278820

This happens to be accurate to within 6 decimals if you check with a calculator. The nice thing about this method is you get about 2 digits of accuracy each time you do an iteration.

Good luck.

*** COMMENT ADDED***
Jeffrey's method below mine works well if you know off hand you only want a few digits of accuracy but the Babylonian method is much better the more digits you need. This is because each calculation you use gives you twice as many digits of accuracy.

Answer 2

There are techniques for calculating square roots in your head. Here's one that I use:


(I've had to use ......... for spacing).


Firstly, I'll show you a process called "cross multiplication" (not the sort you're use to though):

To cross multiply a single digit, you square it.

3² -> 9 for example

To cross multiply two digits, you multiply them and double the answer, an example:

34 -> (3 * 4) * 2 = 24

With three digits, multiply the first and third digits, double the answer, and add this to the square of the middle digit. Example:

345 -> (3*5)*2 + 4² = 46

So a general rule for cross multiplication for an even digit number is:

Multiply the first digit by the last digit, the second by the second last, the third by the third last and so on, until you have multipled all of the digits. Add them together and double the answer.

And a general rule for odd digit number:

Multiply the first digit by the last digit, the second by the second last, the third by the third last and so on, until you have multipled all of the digits up to the middle digit. Add the answers and double the total. Then square the middle digit and add it to the total.

An example:

123 = 1 * 3 = 3, 3 * 2 = 6, 6 + 2² = 10
1234 = 1 * 4, +2*3 = 10, 10 * 2 = 20
...

Now that you know how to use this technique, I'll show you how to apply it to find exact square roots:

An example:

sqrt(2,809) =

Firstly pair the digits:

sqrt(28 09) =
.........*......*

Now estimate the square root of the first digit pair. The square root of 28 is approximately 5. So 5 is the first digit of the answer.

Double the first digit of the answer (2*5=10) and write it to the left of the number. This number will be our divisor. Write 5, the first digit of our answer, above the 8 in the first digit pair, 28.

If you're doing this on paper, it'll look something like:

.............5
10 sqrt(28 09) =

We have finished working on the first digit of the answer.

To find the second digit of the answer, square the first digit of your answer and subtract the answer from your first digit pair.

5² = 25
28 - 25 = 3

Three is our remainder. Carry the 3 remainder to the next digit of the number being squared. This gives us a new working number of 30.

Divide out new working number, 30, by our divisor, 10. This gives 3, the next digit of our answer. Ten divides evenly into 30, so there is no remainder to carry. Nine is our new working number.

Our work should look like this:

.............5....3
10 sqrt(28 09) =
............25...³

Finally, cross multiply the last digit of the answer. (We don't cross multiply the first digit of our answer. After the initial workings the first digit of the answer takes no further part in the calculation.)

3²=9

Subtract this answer from our working number.

9-9 = 0

There is no remainder: 2,809 is a perfect square. The square root is 53.


Go over this, and try and apply it to some other examples. After some practise, you'll learn to do it quickly (almost instantly) in your head.

If you need some more examples, let me know...

Answer 3

if you cant do it on a calculator the simplest way is to use logrithms;

find the log of the number, divide that by two then find the antilog of the result.

there is a modified form of long division that you can use but it would be difficult to explain here. I will try to give you the first couple of steps;

1. place the number under th division sign

2. mark of the digitsd of the number in pairs starting at the decimal point and going left and right.

3. look at the leftmost pair (or single digit if it came out that way and decide what is the biggest number when squared does not exceed this leftmost pair. in your example of 289
the leftmost is 2 so 1 is the number you would choose.

4. write this number in front of the division sign and on top of the division sign

5. multiply the number from the top times the number in front and put the result under the leftmost pair.

6. subtract the number for the above step from the leftmost of number or pair. write this down and bring down the next pair.
from you example subtract the 1 from step 3 from the 2 in 289 and bring down the 89. the number you are looking at for the next step is 189.

7. take the number from above the division sign and double it and place it in fron of 189. in this case put 2 in front. now imagine this 2 is twenty something. what is the largest 20 something that will go into 189? the answer is 7.

8. put the 7 to the right of the one above the divide sign and also to the right of the two in step 7. you will have 17 on top and 27 beside the 189. now multiply the last number that you put on top (7) times the 27 and place the result under 189.
subtract and bring down the next pair as in step 6. in the present case there is no remainder so you have completed the process.

9. if there is a remainder you continue doing steps 6,7 and 8.
add a decimal point and zeros to the original number and continue until you have the accuracy that you need .


I hope you can understand this. its not difficult but it is hard to explain clearly.

Answer 4

Well, the square root of 289 is 17. The best way for most people that I know of is guess and test. For example, if you're trying to find the square root of 1,024, you might try squaring 30, which would give you 900. This is fairly close, but below 1,024, so you know you need to keep going up, but not by a whole lot. You could then try 35, which would give you 1,225. This is above 1,024, so you know that the answer is somewhere between 30 and 35. It's then pretty easy to narrow it down to 32.

Answer 5

Basically square roots is the number before it is squared. So for your example sqrt500 = sqrt (100*5) = 10sqrt (5) An exact answer for sqrt 5 is hard to be calculated with an very exact answer. So you can first use an calculator, complex Taylor series to get an approximation (calculus involved) or trial and error whichbis to multiply 2 same number until you get 5

Answer 6

take the number... lets say 25...and see if there are any numbers that if you multiply them by itself the product is 25....that number is 5.... here are some perfect squares...
1 x 1 = 1
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6= 36
7 x 7 = 49
8 x 8 = 64
9 x 9 = 81
10 x 10 = 100

Answer 7

There is a method similar to doing long division (similar but obviously not the same) that does allow you to compute square roots. You can find the methodology at

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

Answer 8

using a calc. you can buy CASIO sceinific calculator[Model SF500ES] or more less may solve it but beggining from SF82 it beggins to solve well using ur mind sqrt of 289=17 cuz 17*17 =289=17^2

^:power
*:Multiply

Answer 9

use a scientific calculater