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.