Simplify ((8×4)^-7)^6: Nested Negative Exponents Problem

To solve this expression, we will follow these steps using the rules of exponents:

• Step 1: Apply the power of a power rule.
• Step 2: Use the negative exponent rule to express the result as a fraction.

Now, let's apply each step:

Step 1: Apply the power of a power rule
Given: $\left(\left(8\times4\right)^{-7}\right)^6$.
According to the power of a power rule, $(a^m)^n = a^{m \cdot n}$.
So, $\left(\left(8\times4\right)^{-7}\right)^6 = \left(8\times4\right)^{-7 \times 6} = \left(8\times4\right)^{-42}$.

Step 2: Use the negative exponent rule
Now, apply the negative exponent rule: $a^{-m} = \frac{1}{a^m}$.
Thus, $\left(8\times4\right)^{-42} = \frac{1}{\left(8\times4\right)^{42}}$.

The simplified expression is $\frac{1}{\left(8\times4\right)^{42}}$.

Now, let's determine which of the provided answer choices is correct:
- Choice 1: $\frac{1}{\left(8\times4\right)^{-42}}$ is incorrect because the exponent should not be negative.
- Choice 2: $\frac{1}{\left(8\times4\right)^{42}}$ is correct as it matches our solution.
- Choice 3: $\frac{1}{\left(8\times4\right)^{-1}}$ is incorrect because it does not match our calculated exponent.
- Choice 4: $\frac{1}{\left(8\times4\right)^1}$ is incorrect as the exponent is too small.

Therefore, the correct answer is $\frac{1}{\left(8\times4\right)^{42}}$, which corresponds to Choice 2.