Cubed
When a value is followed by the term "cubed," it means that the value is being raised to the power of three. For example, three cubed is written as follows:
In the above figure, "cubed" refers to the exponent, 3.
Cubing a value (raising it to the power of 3) just means to multiply the number by itself two times: 33 = 3 × 3 × 3 = 27. One way to visualize this is to use a cube that is made up of cubes with side lengths of 1 unit (unit cube).
Each of the 6 faces of the cube above has side lengths of 3 units. The total number of unit cubes that make up the whole cube with side lengths of 3 units is 3 x 3 x 3, or 33 = 27. This is one way to view the concept of volume (as the number of unit cubes that can fit in the space).
Perfect cubes
Perfect cubes are the cubes of the integers. Memorizing or at least recognizing perfect cubes can be useful, particularly when solving algebraic equations. Below is a table showing the perfect cubes from 0-12.
| Integer | Squares |
|---|---|
| 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 |
Cubes of negative numbers
The cube of a negative number is always negative. This is because of the nature of negative numbers. A negative value multiplied by another negative is positive. A negative value multiplied by a positive one is negative. In other words, if there are an odd number of negative values being multiplied, the result will be negative. Since a cube involves multiplying the same three numbers, the cube of a negative number will always result in a negative:
(-3)3 = (-3) × (-3) × (-3) = 9 × -3 = -27