Square Root

Posted by Patcharin Sinthunavarat on Saturday, March 24, 2012 Under: Math
 

Squares and Square Roots

First learn about Squares, then Square Roots are easy.

How to Square A Number

To square a number, just multiply it by itself ...

Example: What is 3 squared?

3 Squared = = 3 × 3 = 9







 

 


"Squared" is often written as a little 2 like this:

square root of 9 is 3
This says "4 Squared equals 16"
(the little 2 says the number appears twice in multiplying)


Squares From 12 to 62

1 Squared = 12 = 1 × 1 = 1
2 Squared = 22 = 2 × 2 = 4
3 Squared = 32 = 3 × 3 = 9
4 Squared = 42 = 4 × 4 = 16
5 Squared = 52 = 5 × 5 = 25
6 Squared = 62 = 6 × 6 = 36






 

 

 

You can also find the squares on the Multiplication Table:


Negative Numbers

You 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:

5x5 = -5x-5

(For more detail read Squares and Square Roots in Algebra)

Note: if someone says "minus 5 squared" do you:
  • Square the 5, then do the minus?
  • Or do you square (-5) ?

You get different answers:

Square 5, then do the minus:   Square (-5):
-(5×5) = -25   (-5)×(-5) = +25

 

 

Always make it clear what you mean, and that is what the "( )" are for.


Square Roots

A square root goes the other way:

square root of 9 is 3

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?

tree root To help you remember think of the root of a tree:

"I know the tree, but what is the root that produced it?"

In this case the tree is "9", and the root is "3".


Here are some more squares and square roots:

4   16
5   25
6   36






The Square Root Symbol
 

radical 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!


calculator 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

square root button   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)
around 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 



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