Let's find the Lowest Common Multiple (LCM) of the two numbers $F$ and $G$, given their prime factorizations.

The prime factorization of $F$ is: $F = 3^{10} \times 11^3 \times 13^2$

The prime factorization of $G$ is: $G = 3^5 \times 5^4 \times 7 \times 11^{12} \times 13$

Steps to find the LCM:

1. Identify all distinct prime factors involved in both $F$ and $G$:

   • Prime factors are: $3, 5, 7, 11, 13$

2. Take the highest power of each prime factor from both factorizations:

   • For $3$, the highest power is $3^{10}$.
   • For $5$, the highest power is $5^4$.
   • For $7$, the highest power is $7^1$.
   • For $11$, the highest power is $11^{12}$.
   • For $13$, the highest power is $13^2$.

3. The LCM is the product of all these prime factors raised to their highest powers: $\text{LCM}(F, G) = 3^{10} \times 5^4 \times 7 \times 11^{12} \times 13^2$

Thus, the LCM of $F$ and $G$, expressed as the product of its prime factors in index form, is: $\boxed{3^{10} \times 5^4 \times 7 \times 11^{12} \times 13^2}$

Would you like further details or have any questions?

Here are five related questions to expand on this topic:

1. How do you find the Greatest Common Divisor (GCD) using prime factorization?
2. What is the relationship between the LCM and GCD of two numbers?
3. How can you apply the LCM in solving real-world problems involving time intervals?
4. How do you factorize larger numbers into their prime factors efficiently?
5. What are the methods for determining whether a number is prime?

Tip: Remember, the LCM is always the product of the highest powers of all prime factors involved in both numbers.