To find the Highest Common Factor (HCF) of the given expressions:

1. Expressions in index form:

   • $x = 5 \times 7^3 \times 11^3 \times 13^5$
   • $y = 2 \times 5^5 \times 7^2 \times 11^3$

2. Steps to compute the HCF:

   • The HCF is obtained by taking the lowest power of each common factor from both expressions.

3. Comparison of factors:

   • For $5$: $x$ has $5^1$, $y$ has $5^5$. Take the lowest power: $5^1$.
   • For $7$: $x$ has $7^3$, $y$ has $7^2$. Take the lowest power: $7^2$.
   • For $11$: $x$ has $11^3$, $y$ has $11^3$. Take the lowest power: $11^3$.
   • For $13$: $x$ has $13^5$, $y$ has no $13$. Ignore $13$.

4. HCF in index form: $\text{HCF} = 5^1 \times 7^2 \times 11^3$

5. Final Answer: $\boxed{5 \times 7^2 \times 11^3}$

Would you like further clarification or a step-by-step breakdown? Let me know! 😊

Related Questions:

1. How can you compute the HCF of two numbers given in expanded form instead of index form?
2. What is the relationship between HCF and Least Common Multiple (LCM)?
3. How can HCF be applied in real-world problems involving fractions?
4. How would you handle HCF problems where there are variables (e.g., $x^a$) instead of just constants?
5. What is the role of prime factorization in calculating HCF?

Tip:

When finding HCF, always focus on common prime factors and take the minimum power for each.