Evaluate ((12×5)⁴)⁸: Complex Nested Exponent Problem

To solve this problem, we'll follow these steps:

• Step 1: Identify the given expression

• Step 2: Apply the appropriate exponent rule

• Step 3: Simplify the expression

Now, let's work through each step:
Step 1: The problem gives us the expression $((12 \times 5)^4)^8$. Here, the base is $12 \times 5$, and the exponents are $4$ and $8$ respectively.
Step 2: We'll use the Power of a Power Rule, which states $(a^m)^n = a^{m \times n}$. This rule allows us to combine the exponents by multiplying them together.
Step 3: Applying this rule, we rewrite the expression as:
$((12 \times 5)^4)^8 = (12 \times 5)^{4 \times 8}$

Therefore, the simplified expression is $(12 \times 5)^{32}$.

Now, let's consider the choices provided:

• Choice 1: $\left(12 \times 5\right)^{4 \times 8}$ - This matches our simplified expression.

• Choice 2: $\left(12 \times 5\right)^{8-4}$ - Incorrect because it subtracts exponents rather than multiplying them.

• Choice 3: $\left(12 \times 5\right)^{4+8}$ - Incorrect because it adds exponents rather than multiplying them.

• Choice 4: $\left(12 \times 5\right)^{\frac{8}{4}}$ - Incorrect because it divides exponents rather than multiplying them.

Hence, the correct choice is Choice 1: $(12 \times 5)^{32}$.