Evaluate ((9×3)^-4)^-6: Nested Negative Exponents Problem

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

• Step 1: Identify the expression given: $\left(\left(9 \times 3\right)^{-4}\right)^{-6}$
• Step 2: Apply the power of a power rule
• Step 3: Simplify the expression

Now, let's work through each step:

Step 1: The given expression is $\left(\left(9 \times 3\right)^{-4}\right)^{-6}$.

Step 2: We'll apply the power of a power rule which states that $\left(a^m\right)^n = a^{m \cdot n}$. In our case:

• The base $a$ is $9 \times 3$
• The inner exponent $m$ is $-4$
• The outer exponent $n$ is $-6$

By applying the formula, we get:

$\left(\left(9 \times 3\right)^{-4}\right)^{-6} = \left(9 \times 3\right)^{-4 \times -6}$

Step 3: Calculate the exponent:

$-4 \times -6 = 24$

Thus, the expression simplifies to:

$(9 \times 3)^{24}$

Therefore, the solution to the problem is $\left(9 \times 3\right)^{24}$.

Examining the multiple-choice options, the correct choice is:

• Choice 2: $\left(9\times3\right)^{24}$

Thus, Choice 2 is the correct answer, aligning with our calculated solution.