# Cubes and Cube Roots The topic of cubes and cube roots holds a great significance in mathematics especially for kids in higher grades. The concept of cubes and cube roots is very important to develop the important mathematical skills required for various competitive exams and higher studies. Cubes and cube roots are the inverse operations. The concept of Cube roots is often used in formulas of cubic equations, exponents and to find the volume of geometric objects. In particular, cube roots are often applied in several formulas and physical laws. Cube roots are applied to solve cubic equations and to find the dimensions of three-dimensional objects.

## Introduction to Cubes and Cube Roots

### Definition of Cube

The cube of a number is a number obtained by multiplying the number three thrice to itself. A cube is known as a cube because it is used to represent the volume of a cube. In simple words, when a number is raised to exponent three it is called the cube of that number. For example, the cube of 4 is 64. That means 2 × 2 × 2 = 8, and it can be written as 2 ^3. Similarly, the cube of 3 is 27, and the cube of 5 is 125, and so on. A perfect cube is a number that can be represented as the product of three same or equal integers. For example, 216 is a perfect cube because 6 × 6 × 6 = 216.

### Cube of Numbers and Fractions

To find the cube of a number or a fraction multiply the number or fraction by itself then again multiply the result obtained to the original number or fraction itself. The cube of a positive number is always positive and a negative number is always negative. For example, to find the cube of -2 we have to multiply -2 three times which is (-2) × (-2) × (-2). we will first find the value of (-2) × (-2) which is 4. Now, we will multiply the result again with 2 × (-4) which is -8. Hence, the cube of the negative number ‘-2’ is ‘-8’.

### Definition of Cube Roots

The cube root of a number is multiple of that number which when multiplied thrice gives the original number. It is inverse to the cube of a number and can be obtained by using prime factorization. For example, 125 is the cube of 5, and 5 is the cube root of 125. Unlike square roots, cube roots have negative values, and thus, perfect cubes can also have negative values whereas perfect squares cannot have a negative value.

For example, a cube root of – 64 is -4.

Because 64 is a perfect cube, as 64 = 4 x 4 x 4

## Finding the cube root of a number?

To find the cube root of a number, you have to find a number that when multiplied thrice to itself gives the original number. There are various methods to find the cube root of a number like repeated estimation. We can find the cube roots by using the prime factorization method. Here are the steps to find the cube root of a number using the prime factorization:

Step 1- Calculate the prime factors of the given number by using the Prime Factorization method.

Step 2- Group the factors in three, such that each number of the group is the same.

Step 3- Multiply one factor from each group and the resultant number will be your answer.

If there are factors left that cannot be divided equally into groups of three it implies that the given number is not a perfect cube.

### Example

Prime factorization of 64 is 2x 2 x 2 x 2 x 2 x 2, We can group these prime factors in groups of three each which is { 2,2,2 } and { 2,2,2 }. By multiplying one factor from each group we can find the cube root of 64, i.e., 2 x 2 = 4. The cube root of 64 is 4.

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