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The problem provided involves simplifying the two expressions for \( K \) and \( L \) without using a calculator, as stated by the instructions in French ("sans calculatrice, simplifier au maximum"). Here’s how to approach these problems step-by-step.

### Expression for \( K \):
\[
K = \frac{14 \times 10^7 \times 27 \times 10^{-3}}{21 \times 10^2}
\]
1. Combine the constants:
   \[
   14 \times 27 = 378
   \]
   And simplify the denominator:
   \[
   \frac{378}{21} = 18
   \]
2. Now handle the powers of 10:
   \[
   10^7 \times 10^{-3} = 10^{7 - 3} = 10^4
   \]
   So, we now have:
   \[
   K = \frac{18 \times 10^4}{10^2}
   \]
3. Simplifying further:
   \[
   \frac{18 \times 10^4}{10^2} = 18 \times 10^{4 - 2} = 18 \times 10^2
   \]
   Therefore:
   \[
   K = 18 \times 10^2 = 1800
   \]

### Expression for \( L \):
\[
L = \frac{49 \times 10^{-7} \times 6 \times 10^5}{3 \times 10^4 \times 7 \times 10^{-2}}
\]
1. Combine the constants:
   \[
   49 \times 6 = 294
   \]
   \[
   3 \times 7 = 21
   \]
   Simplify the constant fraction:
   \[
   \frac{294}{21} = 14
   \]
2. Now handle the powers of 10:
   \[
   10^{-7} \times 10^5 = 10^{-7 + 5} = 10^{-2}
   \]
   In the denominator:
   \[
   10^4 \times 10^{-2} = 10^{4 - 2} = 10^2
   \]
   Therefore, the expression simplifies to:
   \[
   L = \frac{14 \times 10^{-2}}{10^2}
   \]
3. Simplifying the powers of 10:
   \[
   \frac{10^{-2}}{10^2} = 10^{-2 - 2} = 10^{-4}
   \]
   So:
   \[
   L = 14 \times 10^{-4}
   \]
   Which can be written as:
   \[
   L = 0.0014
   \]

### Final Answers:
- \( K = 1800 \)
- \( L = 0.0014 \)

Would you like further details or explanations on any part of the process?

Here are five follow-up questions to explore:
1. How do powers of 10 combine during multiplication and division?
2. Why do constants simplify before handling the powers of 10?
3. How could you express the results of \( K \) and \( L \) using scientific notation?
4. What would the result of \( \frac{K}{L} \) be after simplifying both expressions?
5. Can you explain why the powers of 10 are negative or positive in some cases?

**Tip:** Always simplify numerical constants before handling the powers of 10 to make the calculations easier.

Simplify the two expressions for K and L without using a calculator as per the problem in the image.

Learn how to simplify two complex expressions using scientific notation and powers of 10. This problem covers key algebraic techniques for simplifying constants and adjusting powers of 10. Suitable for students in Grades 10-12.

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