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Let's simplify the expression step by step.

We are given:

\[
\frac{(2x^{-1}y^{-5})^3}{(x^{-2}y^{-6})^{-2}}
\]

### Step 1: Simplify the numerator and denominator separately.

**Numerator:**
\[
(2x^{-1}y^{-5})^3
\]

Using the power rule \((a^m)^n = a^{m \cdot n}\), distribute the exponent 3:

\[
= 2^3 \cdot (x^{-1})^3 \cdot (y^{-5})^3
= 8x^{-3}y^{-15}
\]

**Denominator:**
\[
(x^{-2}y^{-6})^{-2}
\]

Again, apply the power rule:

\[
= (x^{-2})^{-2} \cdot (y^{-6})^{-2}
= x^{4}y^{12}
\]

### Step 2: Combine the simplified numerator and denominator.

Now we have:

\[
\frac{8x^{-3}y^{-15}}{x^4 y^{12}}
\]

### Step 3: Simplify the expression by applying the quotient rule \(\frac{a^m}{a^n} = a^{m-n}\).

For the \(x\)-terms:

\[
\frac{x^{-3}}{x^4} = x^{-3 - 4} = x^{-7}
\]

For the \(y\)-terms:

\[
\frac{y^{-15}}{y^{12}} = y^{-15 - 12} = y^{-27}
\]

### Step 4: Final simplified expression.

The expression becomes:

\[
8x^{-7}y^{-27}
\]

Since the problem asks for only positive exponents, rewrite the expression as:

\[
\frac{8}{x^7y^{27}}
\]

### Final Answer:
\[
\frac{8}{x^7 y^{27}}
\]

Would you like more details or have any questions about this process? Here are five related questions to expand on:

1. How does the power of a product rule work in expressions with multiple variables?
2. Why is it necessary to apply the quotient rule when simplifying expressions?
3. How can you convert negative exponents to positive exponents?
4. Can you generalize the rule for simplifying fractional exponents?
5. How would the simplification change if the original expression had different exponents?

**Tip:** When simplifying expressions, always start by applying the power rules to both the numerator and the denominator separately before combining them.

Simplify (2x^(-1) y^(-5))^3 / (x^(-2) y^(-6))^(-2). Write your answer using only positive exponents.

Learn how to simplify the algebraic expression (2x^(-1) y^(-5))^3 / (x^(-2) y^(-6))^(-2) by applying exponent rules. Detailed steps show how to handle negative exponents and write the answer with positive exponents. Suitable for students in Grades 9-12.

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