How to Find Cube Root?

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Cube Root Basics

The cube root of any number is another number which when multiplied by itself twice gives the number whose cube root is to be determined. Cube of root represented using the symbol ∛. Cube root of any number can also be represented in the form of an exponent as the number to the power ⅓. The cube root of any real number, say ‘k’, its cube root can either be written as ∛k or (k). It is very important for us to understand that the cube and cube roots are inverse mathematical operations. So, the cube root of a cube of the number is the number itself. 


How to Find Cube Root?

The cube root of any real number is obtained by either prime factorization method or estimation method when the number whose square root is to be found is a perfect cube number. However, in most cases it is very much recommended to memorize the cubes and cube roots of the first 25 natural numbers at the least. This will help the students and the facilitators to achieve excellent scores in competitive examinations. The table below gives the cubes and cube roots of the first 20 natural numbers. 

Number

Cube 

Cube root

Number

Cube 

Cube root

1

1

1.000

11

1331

2.224

2

8

1.260

12

1728

2.289

3

27

1.442

13

2197

2.351

4

64

1.587

14

2744

2.410

5

125

1.710

15

3375

2.466

6

216

1.817

16

4096

2.520

7

343

1.913

17

4913

2.571

8

512

2.000

18

5832

2.621

9

729

2.080

19

6859

2.668

10

1000

2.154

20

8000

2.714


How to Find Cube Root by Prime Factorization Method?

In this method, the number whose cube root is to be found is resolved completely into its prime factors. The identical prime factors are grouped such that three identical factors form one group. To determine the cube root, one factor from each group is collected and multiplied together. 


Step 1:

The given number is completely resolved into its prime factors. It is always recommended to start the division with the lowest possible prime number and then go to the higher prime number when the quotient is not completely divisible by the number chosen. 

The given number whose cube root is to be determined be 74088. The prime factorization can be summarised as shown in the figure below. 

(Image to be added soon)


Step 2:

Write the number whose cube root is to be determined as the product of their primes. 

The number 74088 can be written as the product of its primes as:

74088 = 2 x 2 x 2 x 3 x 3 x 3 x 7 x 7 x 7


Step 3:

Divide the factors into groups consisting of three identical factors.

74088 as the product of its primes is rewritten as:

74088 = 2 x 2 x 2 x 3 x 3 x 3 x 7 x 7 x 7


Step 4:

The cube root of the number is found as the product of one factor taken from each group in step 3.

So, the cube root of 74088 is found to be 2 x 3 x 7 = 42


How to Find the Cube Root by Estimation Method?

In the estimation method, we segregate the number into groups and then estimate its cube root. The steps to be followed while determining the cube root by estimation method is described below. 


Step 1:

The given number is divided into groups of 3 digits starting from the rightmost digit of the number. If any number is left out without forming a group of three, zeros are appended to its left to make it a group of 3 digits. However, we must take care that the place value of the digit is not altered by appending zeros. 

Let us try finding out the cube root of 74088.

So, to find the cube root of 74088, we should divide the number into groups of three digits starting from the digit in the unit’s place.

074 088


Step 2:

From the first group starting from the right, note down the unit’s digit.

The first rightmost group in the step 1 is 088 and the digit in its unit’s place is 8.


Step 3:

Estimate the digit in the unit’s place of the cube root of the given number using the lookup table given below.

Digit in the Unit’s place of the number

Digit in the Unit’s place of its Cube root

Digit in the Unit’s place of the number

Digit in the Unit’s place of its Cube root

1

1

6

6

8

2

3

7

7

3

2

8

4

4

9

9

5

5

0

0


The digit in the unit’s place obtained in step 1 is ‘8’ and hence the unit’s digit of the cube root of 74088 is also ‘2’.


Step 4:

Now, consider the second group from the right. Check the perfect cube numbers between which this number lies. Suppose the number in this group lies between A3 and B3 and is more closer to B3, then the ten’s digit of its cube root is considered as B. 

In the given number 74088, the second group of 3 digits from the right is 074.

This number lies between two perfect cube numbers 64 and 125 i.e. 43 and 53. Because 74 is more close to 64 i.e. 43, the ten’s digit of the cube root of 74088 is 4.

The cube root of 74088 found using the estimation method is 42.


Fun Facts

  • Cube and cube root are inverse operations and hence cube root of the cube of a number is the number itself.

  • Cube root of a perfect cube number is always a whole number. 

  • Prime factorization and estimation methods can be used to find the cube roots of perfect cubes only. To determine the cube roots of non perfect cubes, there are a few tricks that can be followed. 

FAQ (Frequently Asked Questions)

1. What is Cube and Cube Root?

Cube and cube root are the most important concepts in various mathematical calculations. A Cube of any number is that number which we get when we multiply the number whose cube is to be determined by itself twice. In other words, the cube of a number is the number raised to the power 3. If the number is ‘p’, then its cube is p x p x p or p3. Cube root is the inverse operation of a cube. The Cube root of a number is that number which when multiplied by itself twice gives the number whose cube root is to be determined as the answer.

2. Where Do We Use Cubes and Cube Roots?

Cubes and cube roots are used in several Mathematical and Physical computations. It is quite often used to derive the solutions of cubic equations. To be more specific, the knowledge of cube roots can be used to determine the dimensions of three dimensional objects with specific value. Cubes and cube roots are also used in day to day calculations of Mathematics while learning the concepts such as exponents.