A number is a perfect cube if it is the result of multiplying a whole number by itself twice more, such as 8³ = 512. Factorization can be used to check if a number is a perfect cube by seeing if it can be broken down into three equal factors. Cubing of exponentials involves multiplying by three the exponent of the exponential term.
The number 512 is indeed a perfect cube because it can be expressed as 8³, which is 8 × 8 × 8. A perfect cube is the result of multiplying a whole number by itself three times. To determine if a number is a perfect cube, you can factor the number and see if it can be divided evenly into three equal parts, which would represent the base number multiplied by itself twice more.
For example, to find out if a number like 27 is a perfect cube, we factor it out and discover that 27 = 3 × 3 × 3, which means 27 = 3³. This makes it a perfect cube. Similarly, by checking whether other numbers can be broken down into a factor that repeats three times, we can identify perfect cubes.
Cubing of exponentials is done by cubing the digit term in the usual way and multiplying the exponent of the exponential term by 3. This is related to the multiplication of exponents when dealing with powers of the same base. For instance, (5³)&sup4; results in 5³x4; because you multiply the exponents, yielding 5¹2;.
The probable question may be:
What is the process of cubing exponentials, and how does it relate to multiplying the exponent of the exponential term by 3?
Illustrate the multiplication of exponents in the context of cubing exponentials, using an example such as (5³)⁴.