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The problem you provided involves determining the third expression given the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of three expressions. Let's break it down.

### Problem:
- **HCF** of the three expressions: \( 3^2 \times 11 \)
- **LCM** of the three expressions: \( 3^4 \times 11^5 \)
- Two of the expressions are:
  1. \( 3^2 \times 11^3 \)
  2. \( 3^3 \times 11^5 \)

### Approach:
1. **HCF and LCM relationships**:
   The HCF is determined by the minimum powers of common prime factors, and the LCM is determined by the maximum powers.

2. **Expression 1**: \( 3^2 \times 11^3 \)
3. **Expression 2**: \( 3^3 \times 11^5 \)

To find the remaining expression (Expression 3), we need to consider the prime factor powers. The unknown expression should satisfy the given HCF and LCM conditions.

### Steps:
- **Prime factor 3**:
   - The HCF shows the minimum power of 3 is \( 3^2 \), and the LCM shows the maximum is \( 3^4 \).
   - Expression 1 has \( 3^2 \), and Expression 2 has \( 3^3 \), so Expression 3 must have \( 3^4 \) to meet the LCM condition.

- **Prime factor 11**:
   - The HCF shows the minimum power of 11 is \( 11 \), and the LCM shows the maximum is \( 11^5 \).
   - Expression 1 has \( 11^3 \), and Expression 2 has \( 11^5 \), so Expression 3 must have \( 11 \) to meet the HCF condition.

### Conclusion:
The third expression is:
\[
3^4 \times 11
\]

Would you like a further explanation or any clarifications on this process?

Here are 5 related questions:
1. How do you find the HCF and LCM of algebraic expressions?
2. What is the role of prime factorization in determining the HCF and LCM?
3. Can you explain how the powers of the primes influence the HCF and LCM?
4. How can you check if the remaining expression is correct?
5. How do the properties of exponents help in solving this kind of problem?

**Tip**: Always verify the HCF and LCM by comparing the given expressions with the calculated values for confirmation!

The HCF and LCM of three expressions are 3^2 × 11 and 3^4 × 11^5 respectively. If two of the expressions are 3^2 × 11^3 and 3^3 × 11^5, find the remaining expression.

Learn how to find the third expression when given the HCF and LCM of three algebraic expressions. This problem covers the concepts of prime factorization, HCF, and LCM, and is suitable for students in Grades 9-12.

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