A perfect cube is a number that results from multiplying an integer by itself three times.
Based on the provided definition, what makes a perfect cube a perfect cube is its fundamental property:
Defining a Perfect Cube
According to the definition, a number is known as a perfect cube if it can be decomposed into a product of the same three integers.
Let's break this down:
- Decomposition: This means you can find the factors of the number.
- Product of the same three integers: The number must be the result of multiplying an integer by itself, and then by itself again. For example, if the integer is 'y', the perfect cube 'x' is found by calculating y * y * y.
The Cube Root Connection
The definition also provides an alternative way to understand perfect cubes:
- If the cube root of a number is an integer, then the number can be considered as a perfect cube.
The cube root (represented by the symbol $\sqrt[3]{}$) is the inverse operation of cubing a number. If you take a number, find its cube root, and the result is a whole number (an integer), then the original number is a perfect cube.
As stated in the reference, If x is a perfect cube of y, then we represent it as x = y³. Here, 'y' is the integer cube root, and 'x' is the perfect cube.
Examples of Perfect Cubes
Understanding with examples makes it clearer:
- 1 is a perfect cube because 1 * 1 * 1 = 1. The cube root of 1 is 1 (which is an integer). So, 1 = 1³.
- 8 is a perfect cube because 2 * 2 * 2 = 8. The cube root of 8 is 2 (which is an integer). So, 8 = 2³.
- 27 is a perfect cube because 3 * 3 * 3 = 27. The cube root of 27 is 3 (which is an integer). So, 27 = 3³.
- 125 is a perfect cube because 5 * 5 * 5 = 125. The cube root of 125 is 5 (which is an integer). So, 125 = 5³.
Perfect Cubes vs. Non-Perfect Cubes
To highlight what makes a perfect cube unique, consider numbers that are not perfect cubes:
- 4: You cannot multiply an integer by itself three times to get 4. $\sqrt[3]{4}$ is not an integer.
- 10: You cannot multiply an integer by itself three times to get 10. $\sqrt[3]{10}$ is not an integer.
- 30: You cannot multiply an integer by itself three times to get 30. $\sqrt[3]{30}$ is not an integer.
Summary Table: Cubes of Integers
| Integer (y) | y³ (y * y * y) | Perfect Cube (x) | Cube Root ($\sqrt[3]{x}$) |
|---|---|---|---|
| 1 | 1 * 1 * 1 | 1 | 1 |
| 2 | 2 * 2 * 2 | 8 | 2 |
| 3 | 3 * 3 * 3 | 27 | 3 |
| 4 | 4 * 4 * 4 | 64 | 4 |
| 5 | 5 * 5 * 5 | 125 | 5 |
| ... | ... | ... | ... |
In essence, a number is a perfect cube because it is the third power of an integer. Its defining characteristic, as per the reference, is this specific decomposability into three identical integer factors or, equivalently, having an integer as its cube root.
