Definition of a Perfect Cube
A perfect cube is an integer that results from multiplying an integer by itself three times. In other words, it is the result of cubing an integer (raising it to the power of 3). For instance, 8 is a perfect cube because 2 × 2 × 2 = 8, or 2³.
How to Identify a Perfect Cube
To determine if a number is a perfect cube, you can find its prime factorization. If every prime factor in the factorization appears a number of times that is a multiple of three (e.g., three, six, nine times), then the number is a perfect cube. For example, for 27, the prime factorization is 3 × 3 × 3 = 3³, confirming 27 is a perfect cube.
Examples of Perfect Cubes
The first few positive perfect cubes are 1 (1³), 8 (2³), 27 (3³), 64 (4³), 125 (5³), and so on. Perfect cubes can also be negative; for example, -8 is a perfect cube because (-2)³ = -8.
Applications and Significance
Perfect cubes are fundamental in algebra, particularly when working with cubic equations or expressions. They are also crucial in geometry for calculating the volume of a cube (where V = s³ for side length 's') and play a role in various number theory problems, aiding in the categorization and understanding of integer properties.