We are tasked with simplifying the expression $(4 \times 10^{-5})^{-6}$.

Step-by-Step Solution:

1. Apply the exponent to both factors:

   The expression can be broken into two parts: $(4 \times 10^{-5})^{-6} = 4^{-6} \times (10^{-5})^{-6}$

2. Simplify each part:

   • First, for $4^{-6}$: $4^{-6} = \frac{1}{4^6} = \frac{1}{4096}$

   • Next, for $(10^{-5})^{-6}$: $(10^{-5})^{-6} = 10^{-5 \times (-6)} = 10^{30}$

3. Combine the results: Now, combine both parts: $\frac{1}{4096} \times 10^{30} = \frac{10^{30}}{4096}$

Thus, the simplified form of $(4 \times 10^{-5})^{-6}$ is: $\frac{10^{30}}{4096}$

Let me know if you'd like further details!

---

Here are five related questions to deepen your understanding:

1. How do you simplify expressions with negative exponents?
2. What is the general rule for exponentiation of scientific notation?
3. How do you calculate $4^{-6}$ step-by-step?
4. What is the meaning of scientific notation and when is it used?
5. Can you express $\frac{10^{30}}{4096}$ in decimal form?

Tip: Always apply exponents to each part of the product separately before combining the results.