Let's examine each of the given expressions to determine if prime factorization has been done correctly.
### Expression (a): [tex]\(24 = 2 \times 3 \times 4\)[/tex]
To verify if this is a proper prime factorization:
1. Multiply the factors together: [tex]\(2 \times 3 \times 4 = 24\)[/tex]. The product is correct.
2. Check if all factors are prime numbers:
- [tex]\(2\)[/tex] is a prime number.
- [tex]\(3\)[/tex] is a prime number.
- [tex]\(4\)[/tex] is not a prime number (4 = 2 × 2).
Since [tex]\(4\)[/tex] is not a prime number, this is not a prime factorization.
### Expression (b): [tex]\(56 = 7 \times 2 \times 2 \times 2\)[/tex]
To verify if this is a proper prime factorization:
1. Multiply the factors together: [tex]\(7 \times 2 \times 2 \times 2 = 56\)[/tex]. The product is correct.
2. Check if all factors are prime numbers:
- [tex]\(7\)[/tex] is a prime number.
- [tex]\(2\)[/tex] is a prime number (appears three times, but still valid as each occurrence of 2 is prime).
Since all factors are prime numbers, this is indeed a prime factorization.
### Expression (c): [tex]\(70 = 2 \times 5 \times 7\)[/tex]
To verify if this is a proper prime factorization:
1. Multiply the factors together: [tex]\(2 \times 5 \times 7 = 70\)[/tex]. The product is correct.
2. Check if all factors are prime numbers:
- [tex]\(2\)[/tex] is a prime number.
- [tex]\(5\)[/tex] is a prime number.
- [tex]\(7\)[/tex] is a prime number.
Since all factors are prime numbers, this is indeed a prime factorization.
### Expression (d): [tex]\(54 = 2 \times 3 \times 9\)[/tex]
To verify if this is a proper prime factorization:
1. Multiply the factors together: [tex]\(2 \times 3 \times 9 = 54\)[/tex]. The product is correct.
2. Check if all factors are prime numbers:
- [tex]\(2\)[/tex] is a prime number.
- [tex]\(3\)[/tex] is a prime number.
- [tex]\(9\)[/tex] is not a prime number (9 = 3 × 3).
Since [tex]\(9\)[/tex] is not a prime number, this is not a prime factorization.
### Summary
After analyzing each expression, the following are the valid prime factorizations:
- (b) [tex]\(56 = 7 \times 2 \times 2 \times 2\)[/tex]
- (c) [tex]\(70 = 2 \times 5 \times 7\)[/tex]
Thus, expressions (b) and (c) are the ones where prime factorization has been done correctly.
