Express 900 as a product of its prime factors in index form. Write the prime factors in ascending order.
The correct answer and explanation is:
Correct Answer:
900 = 2² × 3² × 5²
Explanation (300 words):
To express 900 as a product of its prime factors in index form (also known as exponential form), we must first understand what prime factorization means.
A prime factor is a factor of a number that is a prime number—a number greater than 1 that has no divisors other than 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, etc.
Step-by-step Prime Factorization of 900:
- Start by dividing 900 by the smallest prime number (2):
900 ÷ 2 = 450
450 ÷ 2 = 225
Since 225 is not divisible by 2, we move to the next prime. ✅ Prime factor 2 appears 2 times → 2² - Next prime number is 3:
225 ÷ 3 = 75
75 ÷ 3 = 25
25 is not divisible by 3, so we move on. ✅ Prime factor 3 appears 2 times → 3² - Next prime number is 5:
25 ÷ 5 = 5
5 ÷ 5 = 1 ✅ Prime factor 5 appears 2 times → 5²
Now, the division stops when we reach 1. We have fully broken down 900 into its prime components.
Final Step – Writing in Index Form:
Now write the prime factors in ascending order using indices (exponents):
900 = 2² × 3² × 5²
This is the prime factorization of 900 in index form, with all the prime numbers listed in increasing order (2, 3, and 5). This representation is compact, mathematically precise, and shows clearly how many times each prime number appears in the factorization of 900.
This process helps in many mathematical applications, including simplifying fractions, finding greatest common divisors (GCD), least common multiples (LCM), and more.