We are tasked with finding the highest common factor (HCF) of two numbers, $A$ and $B$, which are expressed as products of their prime factors.

From the image, the prime factorizations are given as:

$A = 2^2 \times 3^3 \times 5^2$ $B = 2 \times 3^2 \times 5 \times 7$

Step-by-Step Calculation:

To find the HCF of two numbers, we take the lowest powers of all the common prime factors.

1. Factor 2:

   • In $A$, the power of 2 is $2^2$.
   • In $B$, the power of 2 is $2^1$.
   • The lowest power is $2^1$.

2. Factor 3:

   • In $A$, the power of 3 is $3^3$.
   • In $B$, the power of 3 is $3^2$.
   • The lowest power is $3^2$.

3. Factor 5:

   • In $A$, the power of 5 is $5^2$.
   • In $B$, the power of 5 is $5^1$.
   • The lowest power is $5^1$.

4. Factor 7:

   • $7$ appears only in $B$, so it does not contribute to the HCF.

Putting it together:

The HCF is the product of the lowest powers of the common prime factors:

$HCF = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90$

Thus, the highest common factor (HCF) of $A$ and $B$ is $\boxed{90}$.

Would you like further details or have any questions?

Here are 5 related questions to extend your understanding:

1. How do you find the least common multiple (LCM) using prime factorization?
2. What is the significance of prime factorization in number theory?
3. Can you explain how to find the HCF of more than two numbers?
4. Why is the HCF always a divisor of both numbers?
5. How can the HCF be used in simplifying fractions?

Tip: When finding the HCF, always focus on the common prime factors and choose the lowest power among them.