To write $128$ as a product of its prime factors in index form, we follow these steps:

Step 1: Prime Factorization of $128$

1. $128$ is even, so divide by $2$:
   $128 \div 2 = 64$
2. $64$ is even, so divide by $2$:
   $64 \div 2 = 32$
3. $32$ is even, so divide by $2$:
   $32 \div 2 = 16$
4. $16$ is even, so divide by $2$:
   $16 \div 2 = 8$
5. $8$ is even, so divide by $2$:
   $8 \div 2 = 4$
6. $4$ is even, so divide by $2$:
   $4 \div 2 = 2$
7. Finally, $2 \div 2 = 1$.

Step 2: Write the Result in Index Form

The prime factor $2$ is repeated $7$ times, so: $128 = 2^7$

Final Answer:

The prime factorization of $128$ in index form is: $128 = 2^7$

Would you like a detailed explanation of the process? Here are five related questions:

1. What is the general process of prime factorization?
2. How do you verify if a number is a prime number?
3. Can you explain how index notation works in mathematics?
4. Why is $2$ the only prime factor of $128$?
5. How can this process be extended to non-powers of 2?

Tip: When performing prime factorization, always start with the smallest prime number, $2$, for even numbers!