How to Find Cube Roots Using Factorization – Simple Tips and Tricks
Cube and cube root is one of the most interesting concepts in Mathematics. Whenever a number (x) is multiplied three times, then the resultant number is known as the cube of that number. 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. The cube root of a number can also be exponentially represented as the number to the power 1/3.
Cube root is the factor of a number that is multiplied by itself three times to get the resultant number.
Cube root is represented by the symbol \[ \sqrt[3]{ } \].
For example,
4×4×4= 43 = 64
So,\[ \sqrt[3]{ 64} \] = 4.
The cube root of a number ‘x’ is denoted as \[ \sqrt[3]{x} \] or (x)⅓. 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.
Properties of the Cube
Cubes of odd numbers are odd.
Cubes of even numbers are even.
Cubes of numbers that end with 2 will end in 8. Similarly, cubes of numbers that end with 8 will always end with 2.
Cubes of numbers that end with 3 will end in 7. Similarly, cubes of numbers that end with 7 will always end with 3.
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 the prime factorization method are summarised 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:
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 are 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
\[ \sqrt[3]{ 46656} \] = 2×2×3×3 =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.
The 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 are listed below:
Conclusion
The cube of a number (x) is the resultant number when it is multiplied three times. Basically, the primary source or origin is known as the root. So, all we have to do is consider "which number's cube should be taken to get the given number." The definition of cube root in Mathematics is: It is the number that must be multiplied three times to obtain the original number.
FAQs on Cube Root by Factorization Method: Step-by-Step Guide
1. What is the prime factorization method for finding a cube root?
The prime factorization method is a technique used to find the cube root of a perfect cube number. It involves breaking down the number into its fundamental prime factors. Once the prime factors are listed, they are grouped into identical sets of three, known as triplets. The cube root is then found by multiplying one factor from each of these triplets.
2. What are the key steps to find a cube root using the prime factorization method?
To find the cube root of a number by prime factorization, you should follow these steps as per the CBSE Class 8 syllabus (2025-26):
- Step 1: Find the prime factors of the given number.
- Step 2: Group the identical prime factors into sets of three (triplets).
- Step 3: Take one prime factor from each triplet.
- Step 4: Multiply these selected factors together. The resulting product is the cube root of the original number.
This method only works if the number is a perfect cube, meaning all its prime factors can be successfully grouped into triplets.
3. How can you find the cube root of 512 using the prime factorization method as an example?
To find the cube root of 512, we first find its prime factors: 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. Next, we group these factors into identical triplets: (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2). From each triplet, we take one factor, which gives us 2, 2, and 2. Finally, we multiply these factors: 2 × 2 × 2 = 8. Therefore, the cube root of 512 is 8.
4. Why must prime factors be grouped in triplets to find a cube root?
The concept of a cube root is the inverse of cubing a number. When a number is cubed (e.g., x³), it is multiplied by itself three times (x × x × x). Consequently, the prime factors of this cubed number will consist of three identical sets of the prime factors of 'x'. By grouping the factors into triplets, we are essentially reversing this process to identify the original number 'x' that was multiplied three times to produce the perfect cube.
5. How does the prime factorization method help identify if a number is a perfect cube?
A number is considered a perfect cube if all its prime factors can be grouped into complete triplets with no factors remaining. When you perform prime factorization, if you find that every prime factor appears in multiples of three (e.g., three times, six times, etc.), the number is a perfect cube. If any factor is left over after grouping, the number is not a perfect cube.
6. What is the application of this method for a number that is not a perfect cube, for example, 392?
For numbers that are not perfect cubes, the prime factorization method is used to find the smallest number to multiply or divide by to make it a perfect cube. For example, the prime factors of 392 are 2 × 2 × 2 × 7 × 7. Here, the factor 2 forms a triplet (2 × 2 × 2), but the factor 7 only appears twice. To make it a perfect cube, we need one more 7. Therefore, you must multiply 392 by 7 to make it a perfect cube (392 x 7 = 2744, and the cube root of 2744 is 14).
7. How is finding a cube root by factorization different from finding a square root by the same method?
The core difference lies in the grouping of factors. For finding a square root, prime factors are grouped into identical pairs (sets of two) because a square is a number multiplied by itself twice (x²). For finding a cube root, prime factors are grouped into identical triplets (sets of three) because a cube is a number multiplied by itself three times (x³).
8. What is a limitation of using the prime factorization method for finding cube roots?
While the prime factorization method is very effective for understanding the concept and for solving problems with smaller numbers as per the NCERT syllabus, its main limitation is its inefficiency with very large numbers. Finding the prime factors of a number with many digits or large prime factors can be extremely time-consuming and complex without the help of a calculator.
