To solve the given problem, we simplify the expression step by step. The problem asks to simplify:

$(2^3)^4 \times (2^3)^{-5}$

Step 1: Simplify powers

Using the rule $(a^m)^n = a^{m \cdot n}$, simplify each term: $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$ $(2^3)^{-5} = 2^{3 \cdot (-5)} = 2^{-15}$

Step 2: Combine the terms

Now, use the rule $a^m \cdot a^n = a^{m+n}$ to combine the terms: $2^{12} \times 2^{-15} = 2^{12 + (-15)} = 2^{-3}$

Step 3: Simplify further

The result $2^{-3}$ can be written as: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Final Answer:

The simplified form is: $\boxed{\frac{1}{8}} \quad \text{(Option E).}$

Let me know if you need further clarification or additional steps explained!

Related Questions:

1. How do the laws of exponents simplify expressions with powers?
2. What happens if one of the exponents was zero in this case?
3. How can negative exponents be interpreted with fractions?
4. How can the same principles be applied to expressions with variables instead of numbers?
5. Why do we add exponents when multiplying like bases?

Tip:

Always write the rules of exponents clearly before solving problems, as this ensures accuracy in simplifying complex expressions!