Solve ((6×2)⁴)⁻⁵: Complex Negative Exponent Expression

Insert the corresponding expression:

$\left(\left(6\times2\right)^4\right)^{-5}=$

To solve this problem, we'll simplify the expression $\left(\left(6\times2\right)^4\right)^{-5}$ using exponent rules.

Here's a step-by-step breakdown of the solution:

• Step 1: Identify the form
  The expression is $\left((6 \times 2)^4\right)^{-5}$. This is a case of the power of a power: $(a^m)^n$, which can be rewritten as $a^{m \times n}$.
• Step 2: Apply the power of a power rule
  Apply the rule: $\left((6 \times 2)^4\right)^{-5} = (6 \times 2)^{4 \times (-5)}$.
• Step 3: Simplify the exponent
  Calculate the exponent multiplication: $4 \times (-5) = -20$. Thus, the expression simplifies to $(6 \times 2)^{-20}$.

The resulting expression matches the format of choice 3: $\left(6 \times 2\right)^{4\times-5}$.

Therefore, the correct choice is Choice 3, $\left(6\times2\right)^{4\times-5}$.