Let's perform the prime factorization of 252 and express it in index (exponent) form.

Step-by-Step Prime Factorization:

1. Start by dividing 252 by the smallest prime number, 2: $252 \div 2 = 126$

2. Divide 126 by 2 again: $126 \div 2 = 63$ Since 63 is odd, 2 is no longer a factor.

3. Next, divide 63 by 3 (the next smallest prime): $63 \div 3 = 21$

4. Divide 21 by 3 again: $21 \div 3 = 7$

5. Now, 7 is a prime number, so we stop here.

Prime Decomposition in Index Form:

252 can be expressed as: $252 = 2^2 \times 3^2 \times 7^1$

This is the prime factorization of 252 in index form.

Would you like more details or have any questions? Here are some related questions:

1. How can you verify that the prime decomposition of a number is correct?
2. What are the divisibility rules for 2, 3, and 7?
3. How do you find the prime factorization of a larger number?
4. Why is prime factorization useful in finding the greatest common divisor (GCD)?
5. Can every number be expressed as a unique product of primes?

Tip: Always start with the smallest prime number and proceed step by step to find the prime factorization of any composite number.