find the cube root of the following numbers by prime factorisation 6859
The correct answer and explanation is:
Correct Answer: The cube root of 6859 is 19.
Explanation (Using Prime Factorization Method):
To find the cube root of a number using prime factorization, the number must be broken down into its prime factors. Then, group the prime factors into triples. Each group of three identical prime factors gives one factor of the cube root.
Let’s apply this to 6859:
Step 1: Try dividing by small prime numbers:
- 6859 is odd, so not divisible by 2.
- 6 + 8 + 5 + 9 = 28, which is not divisible by 3, so 6859 is not divisible by 3.
- 6859 does not end in 0 or 5, so not divisible by 5.
- Try 7: 6859 ÷ 7 = 979.857 (not divisible)
- Try 11, 13, and other primes until finding a match.
Now test 19:
- 19 × 19 = 361
- 361 × 19 = 6859
So, 6859 = 19 × 19 × 19 = 19³
This means the prime factorization of 6859 is:
6859 = 19 × 19 × 19
Since the number is expressed as a cube of a single prime factor, the cube root is:
∛6859 = 19
Why Prime Factorization Works:
Cube root by prime factorization is reliable because it breaks down the number into its basic building blocks. If these can be arranged in triplets of equal factors, then the cube root exists as a whole number. Each triplet represents one unit in the cube root.
In this case, 6859 had exactly three factors of 19. When grouped, it gave a perfect cube root of 19. This method works well for perfect cubes and is especially helpful when calculators are not allowed or available.
Understanding cube roots through prime factorization also strengthens number sense and deepens knowledge of exponents, factorization, and roots.