https://file.aiquickdraw.com/mathbot/file/172677689481981ttqncz.jpg

I see that the image contains three mathematical expressions. I'll break them down and help you with the simplifications or explanations of each.

1. **First Expression:**
   \[
   \frac{a^{-1}}{a^{-3}b^{-3}}
   \]
   Simplification:
   - Using the property of exponents: \(\frac{x^m}{x^n} = x^{m-n}\), apply this to both \(a\) and \(b\).
   \[
   \frac{a^{-1}}{a^{-3}b^{-3}} = a^{-1 - (-3)} b^{3} = a^{2}b^{3}
   \]

2. **Second Expression:**
   \[
   \frac{a^2b^2}{a^2 + 2ab + b^2 - 3ab}
   \]
   Simplification:
   - First, simplify the denominator by combining like terms.
   \[
   a^2 + 2ab + b^2 - 3ab = a^2 - ab + b^2
   \]
   So, the expression becomes:
   \[
   \frac{a^2b^2}{a^2 - ab + b^2}
   \]
   No further simplification is possible without factoring, and the denominator is not factorable in a standard way.

3. **Third Expression:**
   \[
   \left(\frac{a^2 - b^2}{ab}\right)^{-1}
   \]
   Simplification:
   - Apply the negative exponent by taking the reciprocal of the fraction.
   \[
   \left(\frac{a^2 - b^2}{ab}\right)^{-1} = \frac{ab}{a^2 - b^2}
   \]
   Now, factor the difference of squares in the denominator:
   \[
   a^2 - b^2 = (a - b)(a + b)
   \]
   So the expression becomes:
   \[
   \frac{ab}{(a - b)(a + b)}
   \]

Would you like further details or explanations on any step? 

Here are five related questions you could explore:
1. What are the general properties of exponents that can simplify complex expressions?
2. How do we handle negative exponents and fractional exponents?
3. What is the significance of factoring expressions like \(a^2 - b^2\)?
4. How do we simplify expressions with multiple terms in both numerator and denominator?
5. What methods are used to recognize when a denominator is factored fully?

**Tip:** When working with exponents, always be careful to apply exponent laws systematically, especially when you have multiple variables involved.

Simplify the following mathematical expressions: 
1. \( \frac{a^{-1}}{a^{-3}b^{-3}} \) 
2. \( \frac{a^2b^2}{a^2 + 2ab + b^2 - 3ab} \) 
3. \( \left(\frac{a^2 - b^2}{ab}\right)^{-1} \)

Learn how to simplify complex algebraic expressions involving exponents and factoring. This problem covers exponential laws, factoring techniques, and algebraic simplifications. Suitable for students in Grades 9-12.

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