express the prime francization of 128 as a product of prime numbers with exponents.

To express the prime factorization of 128 as a product of prime numbers with exponents, we first need to find the prime factors of 128.

128 can be divided by 2 repeatedly:

1. \( 128 \div 2 = 64 \)
2. \( 64 \div 2 = 32 \)
3. \( 32 \div 2 = 16 \)
4. \( 16 \div 2 = 8 \)
5. \( 8 \div 2 = 4 \)
6. \( 4 \div 2 = 2 \)
7. \( 2 \div 2 = 1 \)

Since we divided by 2 a total of 7 times, the prime factorization of 128 is:

\[ 128 = 2^7 \]

So, the prime factorization of 128 as a product of prime numbers with exponents is:

**\( 2^7 \)**.