The expression in the image is:

$\frac{(a^{-1}b^3)^2}{(a^2b^{-3})^3}$

Let’s simplify this step by step.

Step 1: Simplify the numerator

The numerator is $(a^{-1}b^3)^2$, so apply the power to both terms: $(a^{-1})^2 \cdot (b^3)^2 = a^{-2} \cdot b^6$ Thus, the numerator simplifies to $a^{-2}b^6$.

Step 2: Simplify the denominator

The denominator is $(a^2b^{-3})^3$, so apply the exponent to both terms: $(a^2)^3 \cdot (b^{-3})^3 = a^6 \cdot b^{-9}$ Thus, the denominator simplifies to $a^6b^{-9}$.

Step 3: Combine the fraction

Now, the expression becomes: $\frac{a^{-2}b^6}{a^6b^{-9}}$

Step 4: Simplify the fraction

Simplify the powers of $a$ and $b$:

• For $a$, subtract the exponents: $-2 - 6 = -8$, so the result is $a^{-8}$.
• For $b$, subtract the exponents: $6 - (-9) = 6 + 9 = 15$, so the result is $b^{15}$.

Thus, the simplified expression is: $\frac{b^{15}}{a^8} = a^{-8}b^{15}$

Final Answer:

$a^{-8}b^{15}$

Would you like further details or explanations on any step? Here are five related questions:

1. How do we handle negative exponents when simplifying expressions?
2. Can you explain how to distribute powers across multiple terms in parentheses?
3. What is the rule for combining exponents when dividing like bases?
4. How do you convert negative exponents to positive exponents in the final expression?
5. Could this problem be approached differently using logarithmic properties?

Tip: When simplifying fractions with exponents, always subtract the exponents of like bases to combine terms efficiently.