Cube Root by Factorization Method

Basics of Cube and Cube Root

Cube and cube root is one of the most interesting concepts in Mathematics. A Cube of any number can be determined by multiplying the number by itself twice. The cube of a number can also be exponentially represented as the number to the power of 3. The cube root of any real number is that number which when raised to the power of 3 gives the answer equal to the number whose cube root is to be determined. Cube root of a number can also be exponentially represented as the number to the power ⅓.

The cube root of a number ‘x’ is denoted as ∛x or (x). The cube of natural numbers is called the perfect cube numbers. Any perfect cube number will have a cube root equal to a whole number. Cube root of a number can be found either by estimation method or by prime factorization method. However, these two methods are valid only for perfect cube numbers. 


Finding Cube Root of a Number by Prime Factorization Method

Cube root of a number which is a perfect cube can be determined by the Prime factorization method. The name prime factorization method is because the method involves the process of resolution of the number whose cube root is to be found into its prime factors. The steps to be followed in order to find the cube root of a number using prime factorization method is summarized below with an example. 


Step1:

Obtain the number whose cube root is to be found. 

Example: Let us consider a perfect cube number 32768 whose cube root is to be determined. 


Step 2:

Start dividing the number by the lowest possible prime number until it is not completely divisible by that prime number. Once the number is not divisible by the lowest prime number assumed, try with the next higher prime number. Continue the division process till the final number obtained as a quotient is also a prime number. 

Example: 

2

32768

2

16384

2

8192

2

4096

2

2048

2

1024

2

512

2

256

2

128

2

64

2

32

2

16

2

8

2

4


2

Step 3:

Express the number whose square root is to be determined as the product of their primes.

Example:

32768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2


Step 4: 

Every three identical factors is put in groups.

Example:

32768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2


Step 5:

Take one element from each group and find the product. The product thus obtained is equal to the cube root of the number.

Example:

\[\sqrt[3]{32768}\] = 2 x 2 x 2 x 2 x 2

\[\sqrt[3]{32768}\] = 32


Cube Root by Factorization Method Example Problems

1. Find the Cube Root of 46656 Using the Prime Factorization Method.

Solution:

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46656 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3 

\[\sqrt[3]{46656}\] = 2 x 2 x 3 x 3 = 36


2. Find the smallest number by which 243 should be multiplied to get a perfect cube number.

Solution:

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243 = 3 x 3 x 3 x 3 x 3

243 should be multiplied by 3 to make it a perfect cube number. 


Fun Facts about Cube Roots:

  • Cube root of the numbers ending with 1, 8, 7, 4, 5, 6, 3, 2, 9 and 0 may have 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0 respectively in their unit’s place.

  • Cube root of a non perfect cube number cannot be determined by the prime factorization method. 

  • The number of digits in a number and its cube root is:


No. of Digits in the Number 

No. of Digits in its Cube Root

Less than or equal to 3

1

Greater than 3 and less than or equal to 6

2

Greater than 6 and less than or equal to 9

3

Greater than 9 and less than or equal to 12

4

FAQ (Frequently Asked Questions)

1. Explain the Estimation of Cube Root of Perfect Cube Numbers by Grouping.

  • Let us consider a perfect cube number, say 29791. If the cube root of this number is to be determined, it is divided into groups of three digits starting from the rightmost digit. 

  • Considering from the right side, the right most group gives the unit digit of the cube root and the next group gives the tens place of its cube root.

  • Considering the number taken as example, 29791, it grouped as 29 791

  • Since the number in the rightmost group ends with 1, the unit’s place of its cube root is 1.

  • The number in the second group is 29 which lies between 3³ and 4³ i.e. 27 and 64. The ten’s digit of the cube root is considered to be 3 because the value of the group i.e. 29 is more close to 27 i.e. 3³. 

  • Hence the cube root of the given number is 31.

2. What are the Uses of Prime Factorization Method in the Context of Cubes and Cube Roots?

Prime factorization method of finding the cube root of a number is the method in which the given number is resolved into its prime factors. The identical factors are then grouped. One from each group of three identical factors is considered and their product is found to get the cube root of the number. 

Prime factorization method of finding cube roots is valid only if the given number is a perfect cube number. If not, this method can be used to find the least number to be multiplied or divided by the given number to make it a perfect cube.