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Let's solve the given expression \((x^{-3} \cdot y^3)^7\).

### Step-by-Step Solution:

1. **Distribute the exponent to both terms inside the parentheses:**
   Using the power of a product rule, \((a \cdot b)^n = a^n \cdot b^n\), we distribute the exponent 7 to both \(x^{-3}\) and \(y^3\):
   \[
   (x^{-3} \cdot y^3)^7 = x^{-3 \cdot 7} \cdot y^{3 \cdot 7}
   \]
   
2. **Simplify the exponents:**
   Multiply the exponents:
   \[
   x^{-21} \cdot y^{21}
   \]

Thus, the equivalent expression is \(x^{-21} \cdot y^{21}\).

Now, let's check the answer choices:
- **A: \(x^{21} \cdot y^{-21}\)** — This is incorrect because the exponents are flipped.
- **B: \(x^{-21} \cdot y^{21}\)** — This is correct.
- **C: \((xy)^0\)** — This is incorrect because it simplifies to 1, which is not the same as \(x^{-21} \cdot y^{21}\).

### Final Answer:
The correct answer is **B**.

Would you like more details or have any questions?

### Related Questions:
1. How do you apply the power of a power rule in algebra?
2. What is the product of powers rule and how is it used in simplifying expressions?
3. How do negative exponents affect terms in expressions?
4. What are the common mistakes when multiplying powers with the same base?
5. How would the solution change if the exponent outside the parentheses was different?

### Tip:
When simplifying expressions with exponents, remember to apply the exponent to every term inside the parentheses, and pay close attention to negative exponents!

Learn how to simplify the expression (x^-3 · y^3)^7 using the product of powers rule and the power of a product rule. This problem involves basic algebraic exponent rules and is suitable for students in Grades 8-10.

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