How to Find Cube Root by Prime Factorization Method Step by Step with Solved Examples
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 with Detailed Explanation
1. What is cube root by factorization method?
The cube root by factorization method is a technique used to find the cube root of a number by expressing it as a product of its prime factors in groups of three. In this method:
- Find the prime factorization of the given number.
- Group the identical factors into triples.
- Take one factor from each group of three.
2. How do you find the cube root using factorization?
To find the cube root using factorization, first express the number as a product of prime factors and then group them in triples. Follow these steps:
- Step 1: Find the prime factorization of the number.
- Step 2: Arrange the factors into groups of three identical numbers.
- Step 3: Take one number from each group and multiply them.
3. What is the formula for cube root in prime factor form?
The cube root in prime factor form is calculated using the rule: if a number is expressed as a3 × b3, then its cube root is a × b. In general:
- If N = p3a × q3b × r3c,
- Then ∛N = pa × qb × rc.
4. How do you find the cube root of a perfect cube by factorization?
The cube root of a perfect cube is found by grouping its prime factors into sets of three and multiplying one factor from each group. For example, to find ∛216:
- Prime factorization: 216 = 2 × 2 × 2 × 3 × 3 × 3
- Group in triples: (2 × 2 × 2)(3 × 3 × 3)
- Take one from each group: 2 × 3
5. Can you give an example of finding cube root by prime factorization?
Yes, the cube root of 343 can be found using prime factorization. Example:
- 343 = 7 × 7 × 7
- Group the factors: (7 × 7 × 7)
- Take one 7 from the group
6. How do you find the cube root of a negative number by factorization?
The cube root of a negative number is negative, and you can apply factorization to its absolute value first. For example, to find ∛(−125):
- Factorize 125 = 5 × 5 × 5
- Group: (5 × 5 × 5)
- Take one 5 and apply the negative sign
7. What are the steps to check if a number is a perfect cube using factorization?
A number is a perfect cube if all prime factors in its prime factorization appear in multiples of three. To check:
- Find the prime factorization.
- Count the exponent of each prime factor.
- If every exponent is divisible by 3, the number is a perfect cube.
8. What is the difference between cube root by factorization and long division method?
The cube root by factorization method uses prime factors grouped in triples, while the long division method uses a step-by-step division process similar to square root extraction. Key differences:
- Factorization method: Best for perfect cubes, uses prime factorization.
- Long division method: Works for both perfect and non-perfect cubes.
- Factorization is simpler when factors are easy to find.
9. How do you find the smallest number to multiply to make a perfect cube?
To find the smallest number to multiply to make a perfect cube, ensure all prime factor exponents become multiples of three. Steps:
- Find the prime factorization of the number.
- Identify factors whose exponents are not divisible by 3.
- Multiply by the required factors to make each exponent a multiple of 3.
10. What are common mistakes when finding cube root by factorization?
Common mistakes in the cube root by factorization method include incorrect grouping and incomplete prime factorization. Avoid these errors:
- Not finding the complete prime factorization.
- Failing to group factors in exact sets of three.
- Ignoring negative signs in negative numbers.
- Assuming a number is a perfect cube without checking exponents.
