Solve (12^8)^4: Evaluating Nested Exponential Expression

To solve this problem, we will use the Power of a Power rule of exponents, which simplifies expressions where an exponent is raised to another power. The rule is expressed as:

$(a^m)^n = a^{m \cdot n}$

Now, let’s apply this rule to the given problem:

$(12^8)^4$

Step-by-step solution:

• Identify the base and exponents: In this case, the base is 12, with the first exponent being 8 and the second exponent being 4.
• Apply the Power of a Power rule by multiplying the exponents: $(8) \cdot (4) = 32$.
• Replace the original expression with the new exponent: $(12^8)^4 = 12^{32}$.

Therefore, the simplified expression is $\mathbf{12^{32}}$.

Let's compare the answer with the given choices:

• Choice 1: $12^4$ - Incorrect, uses incorrect exponent rule.
• Choice 2: $12^{12}$ - Incorrect, uses incorrect exponent multiplication.
• Choice 3: $12^2$ - Incorrect, unrelated solution.
• Choice 4: $12^{32}$ - Correct, matches our calculation.

Thus, the correct choice is Choice 4: $12^{32}$.

Therefore, the expression $(12^8)^4$ simplifies to $12^{32}$, confirming the correct choice is indeed Choice 4.