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 "3
Cubed" as 3^{3}
(the little ^{3} means
the number appears three times in multiplying)
Cubes From 0^{3} to 6^{3}
0 cubed  =  0^{3}  =  0 × 0 × 0  =  0 
1 cubed  =  1^{3}  =  1 × 1 × 1  =  1 
2 cubed  =  2^{3}  =  2 × 2 × 2  =  8 
3 cubed  =  3^{3}  =  3 × 3 × 3  =  27 
4 cubed  =  4^{3}  =  4 × 4 × 4  =  64 
5 cubed  =  5^{3}  =  5 × 5 × 5  =  125 
6 cubed  =  6^{3}  =  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 cubes and cube roots:
1 
1 

2 
8  
3 
27  
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 we use 5 three times in a multiplication we get 125) ...
... so the cube root of 125 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:
So the cube root of −125 is −5
Perfect Cubes
The Perfect Cubes are the cubes of the whole numbers:
Perfect Cubes 

0  0 
1  1 
2  8 
3  27 
4  64 
5  125 
6  216 
7  343 
8  512 
9  729 
10  1000 
11  1331 
12  1728 
13  2197 
14  2744 
15  3375 
etc... 
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)