To simplify the expression $(a - 4 \cdot b^7)^{-4}$, we will follow the rule of exponents.

Step 1: Apply the exponent

The expression has a negative exponent of $-4$, which can be handled by applying the following rule:

$(x)^{-n} = \frac{1}{x^n}$

Thus, we rewrite the expression as:

$(a - 4 \cdot b^7)^{-4} = \frac{1}{(a - 4 \cdot b^7)^4}$

Final Answer:

$\frac{1}{(a - 4 \cdot b^7)^4}$

This is the fully simplified form of the expression.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you simplify expressions with negative exponents?
2. What is the general rule for raising a binomial to a power?
3. How do you expand $(a - b)^n$ using the binomial theorem?
4. Can you explain how to handle fractional exponents?
5. What is the difference between simplifying $x^{-n}$ and $(x - y)^{-n}$?

Tip: Always remember that a negative exponent in an expression can be rewritten as a reciprocal to eliminate the negative sign.