In which of the following expressions has prime factorization been correctly applied: a) 24 = 2 x 3 x 4, b) 56 = 7 x 2 x 2 x 2, c) 70 = 2 x 5 x 7, d) 54 = 2 x 3 x 9?

To determine which of the given expressions correctly represents prime factorization, we need to examine each one more closely. Prime factorization is the process of expressing a number as the product of its prime numbers.

1. 24 = 2 x 3 x 4: This expression is not in prime factorization form. The number 4 can be further factored into primes, specifically, 4 = 2 x 2. Thus, we can rewrite 24 as:
• 24 = 2 x 3 x 2 x 2 = 2³ x 3
2. 56 = 7 x 2 x 2 x 2: In this case, we have 56 factored into primes correctly. The factors here are 7 (a prime number) and 2 (also a prime number), and they have been multiplied together in their simplest form. We can confirm the prime factorization as:
• 56 = 7 x 2³
3. 70 = 2 x 5 x 7: This expression is also correctly represented in prime factorization. All the factors here (2, 5, and 7) are indeed prime numbers, thus this representation is valid.
• 70 = 2 x 5 x 7
4. 54 = 2 x 3 x 9: Similar to the first example, 9 is not a prime and can be broken down further: 9 = 3 x 3. Thus, 54 can be rewritten as:
• 54 = 2 x 3 x 3 x 3 = 2 x 3³

In conclusion, the correct expressions for prime factorization among the options provided are:

• b) 56 = 7 x 2³
• c) 70 = 2 x 5 x 7

Therefore, both options b and c present valid prime factorizations.