A perfect cube is a number that is the cube (raised to the power of 3) of an integer. For example, 8 is a perfect cube because it can be written as 2^3 (2 x 2 x 2). Some other examples of perfect cubes include 27 (3^3), 64 (4^3), and 125 (5^3).

The equation of a perfect cube:

a^3 = aaa where a is a real number.

Note that not all integers are perfect cubes. For example, 7 is not a perfect cube because there is no integer that, when cubed, will equal 7. In other words, there is no integer “a” such that a^3 = 7.

In addition to being the cube of an integer, a perfect cube can also be expressed as the product of an integer and its square. For example, 8 is not only 2^3, but it is also (2 x 2) x 2.

There are also some mathematical properties associated with perfect cubes. For example, the sum of the digits of a perfect cube will always be divisible by 3 (this property is known as the “digital root” test). Additionally, the last two digits of a perfect cube will always be the same as the last two digits of the number being cubed. For example, the last two digits of 8^3 and 8 are both “64”.

In terms of algebraic equations, solving the equation x^3 = a, where a is a perfect cube, is easier than solving a general cubic equation. since, you can factor out x from the left hand side and get x(x^2)= a.

Perfect cubes are also important in number theory and also used in many combinatorial problems.

You can also check if a number is perfect cube or not by taking cube root of that number and check if it is an integer or not.