Let's analyze and simplify the given expression step-by-step.

Expression:

$\frac{(-3^{-1}x y^3)^{-2} \cdot (2^{-3}x^2y^{-1})^3}{(2^3x^{-3}y)^{-2} \cdot (x^{-1}y^3)^3}$

Step 1: Simplify each term using the exponent rules

We use the following exponent rules:

• $(a^m)^n = a^{m \cdot n}$
• $(ab)^n = a^n \cdot b^n$
• $a^{-n} = \frac{1}{a^n}$

Numerator:

1. First term: $(-3^{-1}xy^3)^{-2}$ $(-3^{-1}xy^3)^{-2} = (-1)^{-2} \cdot (3^{-1})^{-2} \cdot (x^{-2}) \cdot (y^{3})^{-2}$ $= 1 \cdot 3^2 \cdot x^{-2} \cdot y^{-6} = 9x^{-2}y^{-6}.$

2. Second term: $(2^{-3}x^2y^{-1})^3$ $(2^{-3}x^2y^{-1})^3 = (2^{-3})^3 \cdot (x^2)^3 \cdot (y^{-1})^3$ $= 2^{-9} \cdot x^6 \cdot y^{-3}.$

Denominator:

3. First term: $(2^3x^{-3}y)^{-2}$ $(2^3x^{-3}y)^{-2} = (2^3)^{-2} \cdot (x^{-3})^{-2} \cdot (y)^{-2}$ $= 2^{-6} \cdot x^6 \cdot y^{-2}.$

4. Second term: $(x^{-1}y^3)^3$ $(x^{-1}y^3)^3 = (x^{-1})^3 \cdot (y^3)^3$ $= x^{-3} \cdot y^9.$

Step 2: Combine the terms

Numerator:

$9x^{-2}y^{-6} \cdot 2^{-9}x^6y^{-3} = 9 \cdot 2^{-9} \cdot x^{-2+6} \cdot y^{-6-3}$ $= 9 \cdot 2^{-9} \cdot x^4 \cdot y^{-9}.$

Denominator:

$2^{-6}x^6y^{-2} \cdot x^{-3}y^9 = 2^{-6} \cdot x^{6-3} \cdot y^{-2+9}$ $= 2^{-6} \cdot x^3 \cdot y^7.$

Step 3: Simplify the fraction

$\frac{9 \cdot 2^{-9} \cdot x^4 \cdot y^{-9}}{2^{-6} \cdot x^3 \cdot y^7} = 9 \cdot \frac{2^{-9}}{2^{-6}} \cdot x^{4-3} \cdot y^{-9-7}$ $= 9 \cdot 2^{-9+6} \cdot x^{1} \cdot y^{-16}.$ $= 9 \cdot 2^{-3} \cdot x \cdot y^{-16}.$

Step 4: Write the final simplified result

$\frac{9x}{8y^{16}}.$

Let me know if you'd like a further breakdown or have any questions! Here are 5 related questions:

1. What are the rules for simplifying exponents with negative bases?
2. How do you handle fractional exponents in such problems?
3. What happens when terms with the same base but different exponents are multiplied?
4. Can you explain why $a^{-n} = \frac{1}{a^n}$?
5. How can such expressions be applied to practical physics problems?

Tip: When simplifying expressions, work term by term and apply exponent rules systematically.