Solve ((2×4)^-2)^4: Compound Exponent Chain Calculation

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

• Step 1: Identify the expression given and simplify the base.
• Step 2: Apply the Power of a Power Rule for exponents.
• Step 3: Match the result with the provided answer choices.

Let's work through each step:

Step 1: The problem gives us the expression $\left(\left(2\times4\right)^{-2}\right)^4$. First, simplify the base: $2 \times 4$ equals 8, so the expression becomes $\left(8^{-2}\right)^4$.

Step 2: Apply the Power of a Power Rule, which states: $(a^m)^n = a^{m \times n}$. Here, $a = 8$, $m = -2$, and $n = 4$. Calculate $m \times n = -2 \times 4 = -8$.

Therefore, $\left(8^{-2}\right)^4$ simplifies to $8^{-8}$, which can be expressed back in terms of the original base $(2 \times 4)$. So we write it as $\left(2 \times 4\right)^{-8}$.

Step 3: Check the given choices:

• Choice 1: $\left(2\times4\right)^{-2+4}$ represents an exponent of 2; incorrect.
• Choice 2: $\left(2\times4\right)^{\frac{4}{-2}}$ simplifies to -2; incorrect.
• Choice 3: $\left(2\times4\right)^{-2-4}$ simplifies to -6; incorrect.
• Choice 4: $\left(2\times4\right)^{-2\times4} = \left(2\times4\right)^{-8}$; correct.

Thus, the correct choice is Choice 4: $\left(2\times4\right)^{-2\times4}$.