We are given the expression \((uv^2)^{-3} / u^{-2}v^6\). Let's simplify it step by step.

### Step 1: Simplify the numerator \((uv^2)^{-3}\)
Using the property of exponents \((ab)^m = a^m \cdot b^m\), we can break it down as follows:
\[
(uv^2)^{-3} = u^{-3} \cdot (v^2)^{-3}
\]
Now simplify the powers:
\[
v^{2 \cdot (-3)} = v^{-6}
\]
So, the numerator becomes:
\[
u^{-3} \cdot v^{-6}
\]

### Step 2: Rewrite the entire expression
Now substitute the simplified form of the numerator into the original expression:
\[
\frac{u^{-3}v^{-6}}{u^{-2}v^6}
\]

### Step 3: Combine the terms using the quotient rule of exponents
Use the rule \(\frac{a^m}{a^n} = a^{m-n}\) to simplify both the \(u\)-terms and the \(v\)-terms:
\[
\frac{u^{-3}}{u^{-2}} = u^{-3 - (-2)} = u^{-1}
\]
\[
\frac{v^{-6}}{v^6} = v^{-6 - 6} = v^{-12}
\]

### Step 4: Final simplified expression
Thus, the expression simplifies to:
\[
u^{-1} \cdot v^{-12} = \frac{1}{uv^{12}}
\]

### Final Answer:
\[
\frac{1}{uv^{12}}
\]

Let me know if you need further explanation!

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**Related questions**:
1. How would the expression change if the power of \(v^2\) was different?
2. What if the exponents were positive in the original expression?
3. How do negative exponents affect terms in a fraction?
4. Can you simplify \((x^2y^{-1})^{-2} / x^3y^2\)?
5. How do the laws of exponents apply to polynomial expressions?

**Tip**: Always apply the exponent rules carefully, such as distributing exponents and subtracting powers when dividing similar bases!

Learn how to simplify the algebraic expression (uv^2)^-3 / u^-2v^6 using the laws of exponents. This step-by-step guide covers the key rules for simplifying expressions with negative exponents and division of terms. Ideal for students in Grades 9-11.

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