To find the Highest Common Factor (HCF) of two numbers, we need to compare their prime factorizations and take the lowest power of each common prime factor.

The two given numbers are:

1. $2^2 \times 3^1 \times 5^4$
2. $2^3 \times 3^2 \times 7^2$

Step-by-step process:

1. Prime factorization comparison:

   • For $2^2 \times 3^1 \times 5^4$, the prime factors are:

     • $2^2$
     • $3^1$
     • $5^4$

   • For $2^3 \times 3^2 \times 7^2$, the prime factors are:

     • $2^3$
     • $3^2$
     • $7^2$

2. Find common factors:

   • The common prime factors are $2$ and $3$.

3. Take the lowest powers of the common primes:

   • For $2$, the lowest power is $2^2$ (from the first number).
   • For $3$, the lowest power is $3^1$ (from the first number).

4. Calculate the HCF: $\text{HCF} = 2^2 \times 3^1 = 4 \times 3 = 12$

So, the HCF of $2^2 \times 3^1 \times 5^4$ and $2^3 \times 3^2 \times 7^2$ is 12.

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Would you like further explanation or have any questions about this process?

Here are some related questions you can explore:

1. How do you find the LCM (Lowest Common Multiple) of two numbers?
2. How can you use prime factorization to solve HCF and LCM problems?
3. What is the difference between HCF and LCM?
4. How do you apply HCF in simplifying fractions?
5. What is the importance of prime factorization in number theory?

Tip: When dealing with HCF and LCM, always remember to focus on the common prime factors for HCF and all prime factors for LCM.