ELL Math
NOTES
Questions/Main
Ideas
Vocabulary:
square
Name _______________________
Period _______ Date_________
Topic: Square Roots and Cube Roots
CLO: Identify squares and cubes. Label the parts of a radical. Use the graphing
calculator to find the square root and the cube root of a number correctly.
How to Square A Number
To square a number, just multiply it by itself ...
cube
Example: What is 3 squared?
3 Squared
=
=3×3=9
This says "4 Squared equals 16"
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
The squares are also
on the Multiplication Table:
Negative Numbers
We can also square negative numbers.
What happens when we square (−5) ?
Answer:
(−5) × (−5) = 25
When we square a negative number we get a positive result.
Square Roots
A 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 we get 9.
It is like asking:
What can we multiply by itself to get this?
Here are some more squares and square roots:
Decimal Numbers:
4
16
5
25
6
36
It also works for decimal numbers.
Try the sliders below. Note: the numbers here are only shown to 2 decimal places.
Using the sliders (remembering it is only accurate to 2 decimal places):

What is the square root of 8?

What is the square root of 9?

What is the square root of 10?

What is 1 squared?

What is 1.1 squared?

What is 2.6 squared?
Negatives:
We found out before that we can square negative numbers:
Example: (−3) squared
(−3) × (−3) = 9
And of course 3 × 3 = 9 also.
So the square root of 9 could be −3 or +3
Example: What are the square roots of 25?
(−5) × (−5) = 25
5 × 5 = 25
So the square roots of 25 are −5 and +5
The Square Root Symbol
This is the special symbol that means "square root".
It is called the radical.
We use it like this:
we would say "square root of 9 equals 3"
Example: What is √25?
Well, we just happen to know that 25 = 5 × 5, so …
when we multiply 5 by itself (5 × 5) we will get 25.
So the answer is:
√25 = 5
But wait a minute! Can't the square root also be −5? Because (−5) × (−5) = 25 too.


Well the square root of 25 could be −5 or +5.
But when we use the radical symbol √ we only give the positive result.
Example: What is √36 ?
Answer: 6 × 6 = 36, so √36 = 6
Perfect Squares
The perfect squares are the squares of the whole numbers:
Perfect
Squares:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15 etc
1
4
9
16 25 36 49 64 81 100 121 144 169 196 225 ...
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!
Cubes and Cube Roots
To understand cube roots, first we must understand cubes ...
How to Cube A Number
To cube a number, just use it in a multiplication 3 times ...
Example: What is 3 Cubed?
3 Cubed =
= 3 × 3 × 3 = 27
Note: we write down "3 Cubed" as 33
(the little 3 means the number appears three times in multiplying)
Some More Cubes
4 cubed = 43 = 4 × 4 × 4 = 64
5 cubed = 53 = 5 × 5 × 5 = 125
6 cubed = 63 = 6 × 6 × 6 = 216
Cube Root
A cube root goes the other direction:
3 cubed is 27, so the cube root of 27 is 3
3
27
The cube root of a number is ...
... a special value that when cubed gives the original number.
The cube root of 27 is ...
... 3, because when 3 is cubed you get 27.
Here are some more cubes and cube roots:
4
64
5
125
6
216
Example: What is the Cube root of 125?
Well, we just happen to know that 125 = 5 × 5 × 5 (if you use 5 three times in a
multiplication you will get 125) ...
... so the answer is 5
The Cube Root Symbol
This is the special symbol that means "cube root", it is the "radical"
symbol (used for square roots) with a little three to mean cube root.
You can use it like this:
(we say "the cube root of 27 equals 3")
You Can Also Cube Negative Numbers
Have a look at this:
When we cube 5 we get 125:
5 × 5 × 5 = 125
When we cube −5 we get −125:
−5 ×−5 × −5 = −125
So the cube root of −125 is −5
Perfect Cubes
The Perfect Cubes are the cubes of the whole numbers:
Perfect
Cubes:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
8
27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 ...
It is easy to work out the cube root of a perfect cube, but it is really hard to work out
other cube roots.
Example: what is the cube root of 30?
Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3
and 4.
Let's try 3.5:
3.5 × 3.5 × 3.5 = 42.875
Let's try 3.2:
3.2 × 3.2 × 3.2 = 32.768
Let's try 3.1:
3.1 × 3.1 × 3.1 = 29.791
We are getting closer, but very slowly ... at this point, I get out my calculator and it
says:
3.1072325059538588668776624275224...
... but the digits just go on and on, without any pattern. So even the calculator's
answer is only an approximation !
etc