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?
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Squares From 12 to 62
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?
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Here are some more squares and square roots:
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The Square Root Symbol
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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