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Let's work through the two parts of the problem, simplifying each expression and ensuring we express everything using positive exponents.

### (a) \((8r^{-5}s^3t)(4r^7s^{-5}t^{-3})\)

We start by multiplying the coefficients, variables with the same base, and handling the exponents:

1. **Coefficients**: \(8 \times 4 = 32\)

2. **For \(r\)**:
   - \(r^{-5} \times r^7 = r^{-5 + 7} = r^2\)

3. **For \(s\)**:
   - \(s^3 \times s^{-5} = s^{3 + (-5)} = s^{-2}\)
   - To make the exponent positive, express \(s^{-2}\) as \(\frac{1}{s^2}\)

4. **For \(t\)**:
   - \(t^1 \times t^{-3} = t^{1 + (-3)} = t^{-2}\)
   - To make the exponent positive, express \(t^{-2}\) as \(\frac{1}{t^2}\)

Thus, the simplified form of (a) is:
\[
\frac{32r^2}{s^2t^2}
\]

### (b) \(\left(2t^{-1}w\right)^{-3}\left(3t^2w^{-2}\right)^2\)

1. **First Expression**: \(\left(2t^{-1}w\right)^{-3}\)
   - Apply the exponent \(-3\) to each factor:
     - \(2^{-3} = \frac{1}{8}\)
     - \(t^{-1 \times -3} = t^3\)
     - \(w^{-3} = w^{-3}\) (keeps the negative exponent for now)

   So, \(\left(2t^{-1}w\right)^{-3} = \frac{t^3}{8w^3}\).

2. **Second Expression**: \(\left(3t^2w^{-2}\right)^2\)
   - Apply the exponent \(2\) to each factor:
     - \(3^2 = 9\)
     - \(t^{2 \times 2} = t^4\)
     - \(w^{-2 \times 2} = w^{-4}\)

   So, \(\left(3t^2w^{-2}\right)^2 = 9t^4w^{-4}\).

3. **Combine the two results**:
   \[
   \frac{t^3}{8w^3} \times 9t^4w^{-4}
   \]

   - **Coefficients**: \(\frac{1}{8} \times 9 = \frac{9}{8}\)
   - **For \(t\)**: \(t^3 \times t^4 = t^{3+4} = t^7\)
   - **For \(w\)**: \(w^{-3} \times w^{-4} = w^{-7}\)
     - To make the exponent positive, express \(w^{-7}\) as \(\frac{1}{w^7}\)

Thus, the simplified form of (b) is:
\[
\frac{9t^7}{8w^7}
\]

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### Final Answers:
- (a): \(\frac{32r^2}{s^2t^2}\)
- (b): \(\frac{9t^7}{8w^7}\)

Would you like more details on any part of this solution, or do you have any questions?

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Here are 5 follow-up questions you might find interesting:
1. How do you simplify expressions with more than three variables?
2. What happens when multiplying or dividing variables with fractional exponents?
3. How do you deal with negative exponents when combining more complex expressions?
4. What are some common mistakes when simplifying exponent expressions?
5. How can you simplify expressions with radicals alongside exponents?

**Tip**: Always combine like terms before converting negative exponents to positive ones; this makes the simplification process smoother!

Simplify and express the answer using positive exponents only for (a) (8r^(-5)s^3t)(4r^7s^(-5)t^(-3)) and (b) (2t^(-1)w)^(-3)(3t^2w^(-2))^2

Learn how to simplify algebraic expressions like (8r^(-5)s^3t)(4r^7s^(-5)t^(-3)) and (2t^(-1)w)^(-3)(3t^2w^(-2))^2 using exponent laws. This problem covers key concepts such as the product of powers, power of a power, and negative exponent rules. Suitable for students in Grades 9-11.

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