Let's analyze the problem:

We are asked to simplify the expression $(r^{-7})^6$.

Step 1: Apply the Power of a Power Rule

The rule states that $(a^m)^n = a^{m \cdot n}$. Applying this to $(r^{-7})^6$, we get: $r^{-7 \cdot 6} = r^{-42}.$

Step 2: Simplify the Negative Exponent

The negative exponent rule states that $a^{-m} = \frac{1}{a^m}$. Applying this to $r^{-42}$, we get: $r^{-42} = \frac{1}{r^{42}}.$

Final Answer:

The expression equivalent to $(r^{-7})^6$ is $\frac{1}{r^{42}}$.

The correct choice is: $\boxed{\frac{1}{r^{42}}}.$

Would you like further clarification or a detailed explanation?
Here are five related questions for further exploration:

1. What is the power rule for exponents and how does it apply to $(a^m)^n$?
2. How do you simplify expressions involving negative exponents?
3. If $(r^{-7})^6$ were $(r^7)^6$, how would the solution differ?
4. Can you calculate $(r^{-2})^5$ using the same rules?
5. Why is $(r^{-7})^6 \neq -r^6$?

Tip: Always remember to handle negative exponents carefully—they indicate reciprocals, not negative values.