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The given expression is:

\[
\frac{\left( u v^6 \right)}{a b^7} \div \frac{\left( u^{-1} v \right)}{a^{-1} b}
\]

To simplify this, let's follow the steps carefully:

### Step 1: Rewrite the division as multiplication by the reciprocal.
When dividing by a fraction, we multiply by its reciprocal. So, the expression becomes:

\[
\frac{\left( u v^6 \right)}{a b^7} \times \frac{a b}{u^{-1} v}
\]

### Step 2: Simplify each component.
- In the numerator of the second fraction, \( u^{-1} \) is the reciprocal of \( u \), so multiplying by \( u^{-1} \) is like dividing by \( u \).
- We now multiply the numerators and denominators.

\[
= \frac{u v^6 \times a b}{a b^7 \times u^{-1} v}
\]

### Step 3: Combine terms and simplify.
- For the \( u \) terms: \( u \times u^{-1} = u^{1 + 1} = u^2 \).
- For the \( v \) terms: \( v^6 / v = v^{6 - 1} = v^5 \).
- For the \( a \) terms: \( a / a = 1 \).
- For the \( b \) terms: \( b / b^7 = b^{1 - 7} = b^{-6} \).

So, the expression simplifies to:

\[
u^2 v^5 b^{-6}
\]

Finally, we can rewrite this as:

\[
u^2 v^5 \frac{1}{b^6}
\]

Or equivalently:

\[
\frac{u^2 v^5}{b^6}
\]

This is the simplified form of the original expression.

Would you like more details or further clarifications?

Here are some related questions:
1. How do you simplify expressions with negative exponents?
2. What are the rules for dividing powers with the same base?
3. How do you handle fractions involving variables in both the numerator and denominator?
4. Can the order of operations change the result of simplifying a complex expression?
5. What is the impact of reciprocal terms in division?

**Tip**: Always look for common factors when simplifying complex rational expressions—canceling them out makes your work much easier.

Explore how to simplify a complex fractional expression involving variables and exponents. Follow a step-by-step solution to simplify \( \frac{\left( u v^6 \right)}{a b^7} \div \frac{\left( u^{-1} v \right)}{a^{-1} b} \) to \( \frac{u^2 v^5}{b^6} \). Understand the rules of exponents and fraction simplification.

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