Cubes and Cube Roots
To understand cube roots, first you 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.
 |
Note: When you see "root" think "I know the tree, but what is the root that produced it?" In this case the tree is "27", and the cube root is "3". |
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: 
(you would say "the cube root of 27 equals 3")
You Can Also Cube Negative Numbers
Have a look at this:
| If you cube 5 you get 125: | 5 × 5 × 5 = 125 | |
| If you cube -5 you 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:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | etc | |
| Perfect Cubes: | 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 !
(Further reading: these kind of numbers are called surds which are a special type of irrational number)