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The problem provided in the image can be translated as:

Given \( a = 16^9 \times 25^{15} \), and the condition \( k \times 10^n < a < (k+1) \times 10^n \), where both \( k \) and \( n \) are positive integers. If \( 1 \leq k < 10 \), determine the values of \( k \) and \( n \).

### Step-by-step solution:

1. **Calculate \( a \):**
   \[
   a = 16^9 \times 25^{15}
   \]
   We know that:
   \[
   16 = 2^4 \quad \text{and} \quad 25 = 5^2
   \]
   So:
   \[
   16^9 = (2^4)^9 = 2^{36}
   \]
   and
   \[
   25^{15} = (5^2)^{15} = 5^{30}
   \]
   Therefore:
   \[
   a = 2^{36} \times 5^{30}
   \]

2. **Approximate \( a \):**
   To simplify, we express this number in terms of powers of 10. The goal is to estimate \( a \) to match it with a range \( k \times 10^n < a < (k+1) \times 10^n \).

   \( 2^{36} \approx 6.87 \times 10^{10} \) and \( 5^{30} \approx 9.31 \times 10^{20} \). Multiplying them together gives:
   \[
   a \approx (6.87 \times 10^{10}) \times (9.31 \times 10^{20}) \approx 64.01 \times 10^{30}
   \]

3. **Determine \( k \) and \( n \):**
   From the approximation, \( a \approx 6.4 \times 10^{31} \). We are given that \( k \times 10^n < a < (k+1) \times 10^n \) and \( 1 \leq k < 10 \).

   So:
   \[
   6 \times 10^{31} < a \approx 6.4 \times 10^{31} < 7 \times 10^{31}
   \]
   This implies:
   \[
   k = 6 \quad \text{and} \quad n = 31
   \]

### Final Answer:
The values of \( k \) and \( n \) are \( k = 6 \) and \( n = 31 \).

Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:
1. How do we estimate large powers like \( 2^{36} \) and \( 5^{30} \) quickly?
2. What is the role of logarithms in simplifying powers when estimating large numbers?
3. Can you try applying the same approach to a different value of \( a \)?
4. How would the solution change if \( k \) were allowed to be any integer, not just \( 1 \leq k < 10 \)?
5. What are the implications of \( k \times 10^n < a < (k+1) \times 10^n \) in real-world applications?

**Tip:** When dealing with large powers, breaking them down into simpler prime factors (like \( 2^x \), \( 5^y \)) makes estimation easier, especially for powers of 10.

Given a = 16^9 × 25^15, and the condition k × 10^n < a < (k + 1) × 10^n, where both k and n are positive integers. If 1 ≤ k < 10, determine the values of k and n.

Learn how to solve for the values of k and n in the inequality k × 10^n < a < (k + 1) × 10^n, where a = 16^9 × 25^15. This problem involves key mathematical concepts like exponents, estimation, and inequalities. Suitable for high school students in Grades 9-12.

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