Solve ((7×6)^-5)^3: Complex Exponent Expression Evaluation

To solve this problem, we must simplify the expression $\left(\left(7 \times 6\right)^{-5}\right)^3$.

We'll follow these steps:

• Step 1: Apply the power of a power rule.
• Step 2: Simplify the expression into a single exponent.
• Step 3: Convert the negative exponent to a fraction form.

Now, let's work through each step:

Step 1: The expression $\left(7 \times 6\right)^{-5}$ is raised to the power 3. By the power of a power rule, we multiply the exponents:

$\left((7 \times 6)^{-5}\right)^3 = (7 \times 6)^{-5 \times 3} = (7 \times 6)^{-15}$

Step 2: This simplifies the expression to $(7 \times 6)^{-15}$.

Step 3: Since we have a negative exponent, we convert it to a fraction:

$(7 \times 6)^{-15} = \frac{1}{(7 \times 6)^{15}}$

Therefore, the simplified expression is:

$\frac{1}{\left(7 \times 6\right)^{15}}$

Comparing this result with the given choices, the correct answer is:

- Choice 3: $\frac{1}{\left(7 \times 6\right)^{15}}$

The other choices are incorrect because they either have the wrong exponent or incorrectly handle the negative exponent.

Thus, the correct answer to the problem is $\frac{1}{\left(7 \times 6\right)^{15}}$.