Cube RootsCube Roots

To understand cube roots, first you To understand cube roots, first you must understand cubes ...must understand cubes ...

How to Cube A NumberHow to Cube A Number ? ?To To cubecube a number, just use it in a multiplication a number, just use it in a multiplication 3 3

timestimes ... ... 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 CubesSome More Cubes

4 cubed = 43 = 4 × 4 × 4 = 64

5 cubed = 53 = 5 × 5 × 5 = 125

6 cubed = 63 = 6 × 6 × 6 = 216

7 cubed = 73 = 7 × 7 × 7 = 343

8 cubed = 83 = 8 × 8 × 8 = 512

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 ...... the 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".

Cube RootCube Root

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

Here are some more cubes and cube roots: Here are some more cubes and cube roots:

The Cube Root SymbolThe 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:

11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 etcetc

PerfecPerfect t CubesCubes::

11 88 2727 6464 121255

212166

343433

51512 2 729 729 100100

0 0 1331331 1

1721728 8

2192197 7

2742744 4 ……

Cube roots of numbers other than perfect cubes

Example: what is the cube root of 30?

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 !