Square Root
Squares and Square RootsFirst learn about Squares, then Square Roots are easy. How to Square A NumberTo square a number, just multiply it by itself ... Example: What is 3 squared?
 
Squares From 1^{2} to 6^{2}
You can also find the squares on the Multiplication Table:  
Negative NumbersYou can also square negative numbers. Example: What happens when you square (5) ?Answer: (5) × (5) = 25 (because a negative times a negative gives a positive) That was interesting! When you square a negative number you get a positive result. Just the same as if you had squared a positive number: (For more detail read Squares and Square Roots in Algebra) Note: if someone says "minus 5 squared" do you:
You get different answers:
Always make it clear what you mean, and that is what the "( )" are for.  
Square RootsA square root goes the other way: 3 squared is 9, so a square root of 9 is 3
A square root of a number is ... ... a value that can be multiplied by itself to give the original number.A square root of 9 is ... ... 3, because when 3 is multiplied by itself you get 9.It is like asking: What can I multiply by itself to get this?
 
Here are some more squares and square roots:
 
The Square Root Symbol

This is the special symbol that means "square root", it is sort of like a tick, and actually started hundreds of years ago as a dot with a flick upwards. It is called the radical, and always makes math look important! 
You can use it like this:
you would say "square root of 9 equals 3"
Example: What is √25?
Well, we just happen to know that 25 = 5 × 5, so if you multiply 5 by itself (5 × 5)
you will get 25.
So the answer is:
√25 = 5
Example: What is √36 ?
Answer: 6 × 6 = 36, so √36 = 6
Perfect Squares
The perfect squares are the squares of the whole numbers:
1  2  3  4  5  6  7  8  9  10  etc  
Perfect Squares:  1  4  9  16  25  36  49  64  81  100  ... 
Try to remember at least the first 10 of those.
Calculating Square Roots
It is easy to work out the square root of a perfect square, but it is really hard to
work out other square roots.
Example: what is √10?
Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.
 Let's try 3.5: 3.5 × 3.5 = 12.25
 Let's try 3.2: 3.2 × 3.2 = 10.24
 Let's try 3.1: 3.1 × 3.1 = 9.61
 ...
Getting closer to 10, but it will take a long time to get a good answer!
At this point, I get out my calculator and it says:
3.1622776601683793319988935444327 But the digits just go on and on, without any pattern. So even the calculator's answer is only an approximation ! 
Note: numbers like that are called Irrational Numbers, if you want to know more.
The Easiest Way to Calculate a Square Root
Use your calculator's square root button! 
And also use your common sense to make sure you have the right answer.
A Fun Way to Calculate a Square Root
There is a fun method for calculating a square root that gets more and more accurate
each time around:
a) start with a guess (let's guess 4 is the square root of 10)  
b) divide by the guess (10/4 = 2.5) c) add that to the guess (4 + 2.5 = 6.5) d) then divide that result by 2, in other words halve it. (6.5/2 = 3.25) e) now, set that as the new guess, and start at b) again 
 Our first attempt got us from 4 to 3.25
 Going again (b to e) gets us: 3.163
 Going again (b to e) gets us: 3.1623
And so, after 3 times around the answer is 3.1623, which is pretty good, because:
3.1623 x 3.1623 = 10.00014
Now ... why don't you try calculating the square root of 2 this way?
How to Guess
What if you have to guess the square root for a difficult number such as "82,163" ... ?
In that case I would think to myself "82,163" has 5 digits, so the square root might
have 3 digits (100x100=10,000), and the square root of 8 (the first digit) is
about 3 (3x3=9), so 300 would be a good start.
A List Of The First 100 Square Numbers  
1 x 1 = 1  21 x 21 = 441  41 x 41 = 1,681  61 x 61 = 3,721  81 x 81 = 6,561 
2 x 2 = 4  22 x 22 = 484  42 x 42 = 1,764  62 x 62 = 3,844  82 x 82 = 6,724 
3 x 3 = 9  23 x 23 = 529  43 x 43 = 1,849  63 x 63 = 3,969  83 x 83 = 6,889 
4 x 4 = 16  24 x 24 = 576  44 x 44 = 1,936  64 x 64 = 4,096  84 x 84 = 7,056 
5 x 5 = 25  25 x 25 = 625  45 x 45 = 2,025  65 x 65 = 4,225  85 x 85 = 7,225 
6 x 6 = 36  26 x 26 = 676  46 x 46 = 2,116  66 x 66 = 4,356  86 x 86 = 7,396 
7 x 7 = 49  27 x 27 = 729  47 x 47 = 2,209  67 x 67 = 4,489  87 x 87 = 7,569 
8 x 8 = 64  28 x 28 = 784  48 x 48 = 2,304  68 x 68 = 4,624  88 x 88 = 7,744 
9 x 9 = 81  29 x 29 = 841  49 x 49 = 2,401  69 x 69 = 4,761  89 x 89 = 7,921 
10 x 10 = 100  30 x 30 = 900  50 x 50 = 2,500  70 x 70 = 4,900  90 x 90 = 8,100 
11 x 11 = 121  31 x 31 = 961  51 x 51 = 2,601  71 x 71 = 5,041  91 x 91 = 8,281 
12 x 12 = 144  32 x 32 = 1,024  52 x 52 = 2,704  72 x 72 = 5,184  92 x 92 = 8,464 
13 x 13 = 169  33 x 33 = 1,089  53 x 53 = 2,809  73 x 73 = 5,329  93 x 93 = 8,649 
14 x 14 = 196  34 x 34 = 1,156  54 x 54 = 2,916  74 x 74 = 5,476  94 x 94 = 8,836 
15 x 15 = 225  35 x 35 = 1,225  55 x 55 = 3,025  75 x 75 = 5,625  95 x 95 = 9,025 
16 x 16 = 256  36 x 36 = 1,296  56 x 56 = 3,136  76 x 76 = 5,776  96 x 96 = 9,216 
17 x 17 = 289  37 x 37 = 1,369  57 x 57 = 3,249  77 x 77 = 5,929  97 x 97 = 9,409 
18 x 18 = 324  38 x 38 = 1,444  58 x 58 = 3,364  78 x 78 = 6,084  98 x 98 = 9,604 
19 x 19 = 361  39 x 39 = 1,521  59 x 59 = 3,481  79 x 79 = 6,241  99 x 99 = 9,801 
20 x 20 = 400  40 x 40 = 1,600  60 x 60 = 3,600  80 x 80 = 6,400  100 x100 = 10,000 
In : Math