Cube root by Prime Factorisation method – Definition & Examples
Cube Root by Factorisation Method
In this lesson we will learn prime factorisation of the numbers and their cubes.
Before we study how to calculate cube root by prime factorisation we will see,
What is prime number, prime factorisation, cube root and cube numbers.
A prime number is a natural number greater than 1, with exactly two factors. A prime number is only divisible by 1 and itself.
The numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, are first few prime numbers.
The numbers that have more than two factors are called composite numbers.
The number 1 is neither prime nor composite.
To find the cube root of a number the only prime factor method is available for solving.
Before learn cube root by prime factorisation method, first we learn meaning of cube and cube root.
When a number is multiplied by itself and again multiplied by itself then the number is known as a cube number.
The power written for cube is 3.
Example: x3
3³ = 3 x 3 x 3 = 27
4³ = 4 x 4 x 4 = 64
There are some steps to solve cube root by prime factorisation method.
Step 1: Take the given number.
Step 2: Find the prime factors of given number.
Step 3: Grouping numbers in such a way that one group of three same prime factors.
Step 4: Collect one prime factor from each group.
Step 5: Multiply one pair of the collected prime factors.
Step 6: This multiplication is required cube root.
Example 1: Find the cube root of 125.
Solution: Steps are as follows
Step 1. Take the given number
The number is 125
Step 2. Find the prime factors of given number.
prime factors of 125 = 5 x 5 x 5
Step 3. Grouping numbers in such a way that one group of three same prime factors.
= 5 x 5 x 5
Step 4. Collect one prime factor from each group.
= 5
Step 5. Multiply one pair of the collected prime factors.
Step 6. This multiplication is required cube root.
= 5
Solution: 5 is the cube root of 125.
Or we can write as
∛125 = 5
Example 2: Find the cube root of 64.
Solution- Steps are as follows
Step 1. Take the given number
The number is 64
Step 2. Find the prime factors of given number.
prime factors of 64 = 4 x 4 x 4
Step 3. Grouping numbers in such a way that one group of three same prime factors.
= 4 x 4 x 4
Step 4. Collect one prime factor from each group.
= 4
Step 5. Multiply one pair of the collected prime factors.
Step 6. This multiplication is required cube root.
= 4
Solution: 4 is the cube root of 64.
Or we can write as
∛64 = 4
Example 3: Find the cube root of 729.
Solution: Steps are as follows
Step 1. Take the given number
The number is 729
Step 2. Find the prime factors of given number.
prime factors of 729 = 9 x 9 x 9
Step 3. Grouping numbers in such a way that one group of three same prime factors.
= 9 x 9 x 9
Step 4. Collect one prime factor from each group.
= 9
Step 5. Multiply one pair of the collected prime factors.
Step 6. This multiplication is required cube root.
= 9
Solution: 9 is the cube root of 729
Or we can write as
∛729 = 9
Example 4: Find cube root of 1000.
We find the cube root by prime factorisation,
Factors of 1000 is
2 x 2 x 2 x 5 x 5 x 5
(2 x 2 x 2) x (5 x 5 x 5)
Each factor comes 3 times.
so, 1000 = 23 x 53
= (2 x 5)3
=(10)3
2 x 5
Therefore, cube root of 1000 = ∛1000
Example 5. Find cube root of 512
Prime factors of 512 is,
Factors of 512 is
(2 x 2 x 2) x (2 x 2 x 2) x (2 x 2 x 2)
Each factor comes 3 times.
so, 512 = 23 x 23 x 23
= (2 x 2
= (8)³
Therefore, cube root of 512 = ∛512
= 2 x 2 x 2
= 8
Example 6: Find the cube root of 1728.
Solution: First we do prime factorisation of 1728
prime factorisation of 1728 =
= 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3
= (2 x 2 x 3) x (2 x 2 x 3) x (2 x 2 x 3)
= (2 x 2 x 3)3
1728 = (12)3
1728 = 12 x 12 x 12
∛1728 = ∛12 x 12 x 12
∛1728 = 12
∴ The cube root of 1728 is 12
Example 7: Find the cube root of 13824.
Solution: First we do prime factorisation of 13824
prime factorisation of 13824 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3
13824 = (2 x 2 x 2) x (2 x 2 x 2) x (2 x 2 x 2) x (3 x 3 x 3)
13824 = (2 x 2 x 2 x 3)³
13824 = 243
∛13824 = ∛24 x 24 x 24
∛13824 = 24
∴ The cube root of 13824 is 24
Example 8: Find the cube root of 3375.
Solution: First we do prime factorisation of 3375
= 3 x 3 x 3 x 5 x 5 x 5
= (3 x 3 x 3) x (5 x 5 x 5)
= (3 x 5)3
3375 = (15)3
3375 = 15 x 15 x 15
∛3375 = ∛15 x 15 x 15
∛3375 = 15
∴ The cube root of 3375 is 15.
Example 9: Find the cube root of 9261
Solution: First we do prime factorisation of 9261
prime factorisation of 9261 = 3 x 3 x 3 x 7 x 7 x 7
9261 = (3 x 7 ) x (3 x 7) x (3 x 7)
9261 = (3 x 7)3
9261 = 213
9261 = 21 x 21 x 21
∛9261 = ∛21 x 21 x 21
∛9261 = 21
∴ The cube root of 9261 is 21
Example 10: Find the cube root of 10648.
Solution: First we do prime factorisation of 10648
prime factorisation of 10648 =
prime factorisation of 10648 = 2 x 2 x 2 x 11 x 11 x 11
10648 = (2 x 2 x 2) x (11 x 11 x 11)
10648 = (2 x 11)3
10648 = (22)3
10648 = 22 x 22 x 22
10648 = (22)3
∛10648 = ∛22 x 22 x 22
∛10648 = 22
∴ The cube root of 10648 is 22
Example 11: Find the cube root of 74088
Solution: First we do prime factorisation of 74088
prime factorisation of 74088 =
= 2 x 2 x 2 x 3 x 3 x 3 x 7 x 7 x 7
= (2 x 3 x 7) x (2 x 3 x 7) x (2 x 3 x7)
= (2 x 3 x 7)3
74088 = (42)3
74088 = 42 x 42 x 42
∛74088 = ∛42 x 42 x 42
∛74088 = 42
∴ The cube root of 74088 is 42
Example 12: Find cube root of 6859
We find the cube root by prime factorisation,
prime factorisation of 6859 = 19 x 19 x 19
6859 = (19 x 19 x 19)
6859 = (19)3
∛6859 = ∛19 x 19 x 19
∛6859 = 19
∴ The cube root of 6859 is 19
Example 13: Find cube root of 256
We find the cube root by prime factorisation,
Prime factors of 256 is,
2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
Each factor not comes 3 times.
256 = 23 x 23 x 22
Therefore, 256 is not a perfect cube.