The given task asks to express the following mathematical expression as a power with a base of 4, and then calculate its value.

The expression is: $\frac{1}{64} \cdot (2^{-2})^{-12} \cdot 16^0 \cdot 256^{-1}$

Step-by-step breakdown:

1. Simplifying each part of the expression:

   • $\frac{1}{64} = 64^{-1} = (4^3)^{-1} = 4^{-3}$
   • $2^{-2} = 4^{-1}$ (since $2^2 = 4$, we can write it as a power of 4)
   • $(2^{-2})^{-12} = (4^{-1})^{-12} = 4^{12}$
   • $16^0 = 1$ (anything to the power of 0 is 1)
   • $256^{-1} = (4^4)^{-1} = 4^{-4}$

2. Combining everything: $4^{-3} \cdot 4^{12} \cdot 1 \cdot 4^{-4}$

3. Simplifying the powers of 4: Using the rule of exponents $a^m \cdot a^n = a^{m+n}$, we combine: $4^{-3 + 12 - 4} = 4^{5}$

4. Calculating the value: $4^5 = 1024$

Final Answer:

The expression simplifies to $4^5 = 1024$.

Would you like more details or have any questions? Here are some related questions to explore:

1. How do we simplify expressions with different bases to a common base?
2. What are the rules for manipulating exponents with negative powers?
3. How do zero powers affect the result in an expression?
4. What are the properties of fractional powers?
5. How can we express numbers like 256 or 16 as powers of different bases?

Tip: When working with exponents, always try to rewrite numbers as powers of a common base to simplify operations.