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The problem asks to find which exponential expression is equal to \((6^{-3})^6\).

Let's simplify \((6^{-3})^6\):

1. Using the power of a power rule in exponents, \((a^m)^n = a^{m \cdot n}\), we multiply the exponents.
   
   \[
   (6^{-3})^6 = 6^{-3 \cdot 6} = 6^{-18}
   \]

Now, we can look at the answer choices:

- **A**: \(\frac{6^3}{6^6}\)

   Applying the quotient rule for exponents \(\frac{a^m}{a^n} = a^{m-n}\):

   \[
   \frac{6^3}{6^6} = 6^{3-6} = 6^{-3}
   \]

   This does not match \(6^{-18}\).

- **B**: \((6^4)^{11}\)

   Apply the power of a power rule:

   \[
   (6^4)^{11} = 6^{4 \cdot 11} = 6^{44}
   \]

   This does not match \(6^{-18}\).

- **C**: \(\frac{6^{-3}}{6^{15}}\)

   Again, apply the quotient rule:

   \[
   \frac{6^{-3}}{6^{15}} = 6^{-3 - 15} = 6^{-18}
   \]

   This matches \(6^{-18}\).

- **D**: \((6^4)^{-5}\)

   Apply the power of a power rule:

   \[
   (6^4)^{-5} = 6^{4 \cdot (-5)} = 6^{-20}
   \]

   This does not match \(6^{-18}\).

Thus, the correct answer is **C**: \(\frac{6^{-3}}{6^{15}}\).

Would you like more details or explanations on any part of the process? Here are 5 related questions to practice further:

1. Simplify \((4^{-2})^5\).
2. What is the simplified form of \(\frac{7^5}{7^{12}}\)?
3. Simplify the expression \((2^3)^4 \times 2^{-6}\).
4. Evaluate \((8^{-1})^3\).
5. Simplify \(\frac{10^{-4}}{10^{-9}}\).

**Tip**: When simplifying exponents, always remember to apply the rules for powers and quotients carefully to avoid mistakes!

Which exponential expression is equal to (6^-3)^6?

Learn how to simplify the exponential expression (6^-3)^6 using power of a power and quotient rules. This problem helps reinforce key concepts in exponent manipulation and is suitable for students in Grades 8-10.

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