Final answer:
To express 75 as a product of prime numbers, we find the smallest prime that divides it, which is 3, then divide the result (25) by 5 to get another 5, leading to the prime factorization of 75 = 3 × 5 × 5 or 75 = 3 × 5².
Step-by-step explanation:
How to Express 75 as the Product of Prime Numbers
To express the number 75 as a product of prime numbers, also known as its prime factorization, we break it down into factors that are all prime numbers. This process is a fundamental concept in Mathematics that involves prime numbers and the product rule. Here's the step-by-step breakdown:
- Start with the smallest prime number that divides 75, which is 3. Dividing 75 by 3 gives us 25.
- Since 25 is not a prime number, we continue factoring it. 25 can be divided by 5, which is a prime number. This gives us 5 as a result.
- Now we have another 5, because 5 × 5 equals to 25.
- So, we can write 75 as a product of its prime factors: 75 = 3 × 5 × 5. Using exponents, this can also be written as 75 = 3 × 5².
To represent larger numbers or exponents compactly, mathematicians use the product rule, which is a way to write repeated multiplication of the same number quickly and efficiently.