Understanding Cube Roots
Finding the cube root of a number means identifying a value that, when multiplied by itself three times, yields that number. For example, if x is the cube root of 15625, then x * x * x = 15625.
Step-by-Step Calculation
1. Prime Factorization
- Start by breaking down 15625 into its prime factors.
- Divide by the smallest prime numbers:
- 15625 ÷ 5 = 3125
- 3125 ÷ 5 = 625
- 625 ÷ 5 = 125
- 125 ÷ 5 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
- The prime factorization is: 15625 = 5^6.
2. Applying the Cube Root Formula
- The cube root can be calculated using the exponent rule:
- The cube root of a number with a prime factorization of p^n is p^(n/3).
- Here, n is 6, so:
- Cube root of 15625 = 5^(6/3) = 5^2.
3. Calculating the Final Value
- Thus, 5^2 = 25.
- Therefore, the cube root of 15625 is 25.
Conclusion
The cube root of 15625 is 25. Understanding prime factorization and applying the cube root formula are essential steps in solving such problems. This method ensures accuracy and clarity in calculations, especially useful for competitive exams like UPSC.