How To Find Cube Root: Prime Factorization and Estimation - Class 9

Let us learn how to find cube root in this blog with the help of examples.

As in squares and square roots we learned that the inverse of the square is the square root similarly inverse of the cube is the cube root. So we will discuss how to find cube root.

There are 2 ways to find the cube root of a number.

These are:
1. Prime factorization
2. Estimation (using unit�s digits)                          

Cube Root Formula: Cube roots are denoted by ³?.

How To Find Cube Root By Prime Factorization Method?

Prime factorization

As the name suggests we will factorize the number using prime numbers as we did for finding square roots. Let us take a cube number say 216.

First, find the prime factors of it.

i.e., 216 = 2 � 2 � 2 � 3 � 3 � 3

Now take one number from each group.
2 � 3

Multiply them.
2 � 3 = 6

Obtained number i.e., 6 is the cube root of the number. So 6 is the cube root of 216 which will be represented as ³?216 = 6

How To Find Cube Root Using Estimation Method?

Estimation (using unit�s digits)

Now we will find the cube root by estimation.

Let us take a number say 3375.

3375 is an odd number. So its cube root will also be odd.

Group the digits in 3 starting from the unit�s place. 3 2nd group, 75 1st group. Using the unit digit of the 1st group we will decide on the unit digit of the cube root. Here unit digit is 5. So the unit digits of the cube root will also be 5.

Think of the cube number as smaller than the number in the second group. In this case, the cube number smaller than the number 3 is 1. So the cube root of 3375 is 15.

2. Find the cube root of 17576 through estimation.

Solution: The given number is 17576.
1. Form groups of three starting from the rightmost digit of 17576.
17 576. In this case, one group i.e., 576 has three digits whereas 17 has only
two digits.
2. Take 576.
The digit 6 is at its one�s place.
We take the one�s place of the required cube root as 6.
3. Take the other group, i.e., 17.
Cube of 2 is 8 and cube of 3 is 27. 17 lies between 8 and 27.
The smaller number between 2 and 3 is 2.
The one�s place of 2 is 2 itself. Take 2 as ten�s place of the cube root of 17576.
Thus, ³?17576 = 26

How to make a perfect cube?

There is an egg rack. here are 6 shelves in a rack. Each shelf has the capacity of 6 x 6 eggs. There are 108 eggs in the rack. How many eggs will we need to make it a perfect cube?

Let us take 108 and find its cube root by prime factorization.

We have: 108 = 2 � 2 � 3 � 3 � 3.

This is not a perfect cube. To make it a perfect cube factors can be grouped in triples. There are 3 three�s in the product but only two 2�s.

Since there are two 2�s in prime factorization, so we need only one more 2 to make it a perfect cube.

108 � 2 = 2 � 2 � 2 � 3 � 3 � 3

So, 216 is a perfect cube number.

So we need 108 eggs to make the number of eggs a perfect cube.

Interesting patterns of cubes

Now, we discuss some interesting patterns of cubes.

See this pattern.

Odd Numbers	Sum of odd numbers	Cube numbers
1	1	13
3 + 5	8	23
7 + 9 + 11	27	33
13 + 15 + 17 + 19	64	43
21 + 23 + 25 + 27 + 29	125	53

The first odd number is the cube of 1 and the sum of the next two odd numbers is the cube of 2 and as before the sum of the next three consecutive odd numbers is the cube of 3 and so on. So, the sum of n odd natural numbers represents the cube of n.

This is the second pattern of cubes and their prime factors.

Look at this table the prime factorization of 14 is 2 and 7, and the cube of 14 is 2744. 2 and 7 appear 3 times in the prime factorization of 2744. Each prime factor appears 3 times in the cube of numbers.

Prime factorization of a number	Prime factorisation of its cube
8 = 2 � 2 � 2	83 = 512 = 2 � 2 � 2 � 2 � 2 � 2 � 2 � 2 � 2 = 23 � 23 � 23
14 = 2 � 7	143 = 2744 = 2 � 2 � 2 � 7 � 7 � 7 = 23 � 73
15 = 3 � 5	153 = 3375 = 3 � 3 � 3 � 5 � 5 � 5 = 33 � 53

Consider the following pattern.

The difference between the cubes of two consecutive numbers can be represented as.

2³ � 1³ = 1 + 2 � 1 � 3

3³ � 2³ = 1 + 3 � 2 � 3

4³ � 3³ = 1 + 4 � 3 � 3

Read More:
Cube and Cube Roots: What is a Cube? Cube Properties With Examples