The given expression is:

$8^8 \cdot 4^2 \cdot 32^7 \quad \text{(use 2 as the base)}$

Step-by-step Solution:

1. Convert each term to base 2:

   • $8 = 2^3$, so $8^8 = (2^3)^8 = 2^{24}$.
   • $4 = 2^2$, so $4^2 = (2^2)^2 = 2^4$.
   • $32 = 2^5$, so $32^7 = (2^5)^7 = 2^{35}$.

2. Combine the exponents: Using the property of exponents $a^m \cdot a^n = a^{m+n}$, we add the exponents: $2^{24} \cdot 2^4 \cdot 2^{35} = 2^{24 + 4 + 35} = 2^{63}$

Final Answer:

$2^{63}$

Would you like more details or further clarifications?

Here are 5 related questions for deeper exploration:

1. How do you simplify expressions using exponent rules?
2. What is the rule for multiplying numbers with the same base?
3. How do you convert powers of integers to powers of 2?
4. What is the general approach for solving exponential equations?
5. How does changing the base affect the result of exponentiation?

Tip: Always remember to break down numbers into their prime factors when working with exponent rules, especially when simplifying complex expressions!