https://file.aiquickdraw.com/mathbot/file/17289052406429ho4kqee.jpg

Let's solve the two parts of the question from the image step by step, simplifying each expression with positive indices.

### Part (a):
\[
\left(\frac{3s^5 t^3}{u^{-7}}\right)^{-2} \times \left(\frac{c^4 d^{-5}}{c^{-7} d}\right)
\]

1. **First, simplify the first fraction raised to the power of \(-2\):**
   Apply the rule \(\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}\).

   \[
   \left(\frac{3s^5 t^3}{u^{-7}}\right)^{-2} = \frac{u^{7 \times 2}}{3^{-2} s^{5 \times (-2)} t^{3 \times (-2)}} = \frac{u^{14}}{3^{-2} s^{-10} t^{-6}}
   \]
   Simplify \(3^{-2}\) to \(\frac{1}{9}\), so:
   \[
   = \frac{u^{14}}{9 s^{-10} t^{-6}} = \frac{u^{14} s^{10} t^6}{9}
   \]

2. **Now, simplify the second fraction:**
   \[
   \frac{c^4 d^{-5}}{c^{-7} d} = c^{4 - (-7)} \times d^{-5 - 1} = c^{4 + 7} \times d^{-6} = c^{11} \times d^{-6}
   \]

3. **Combine both results:**
   \[
   \frac{u^{14} s^{10} t^6}{9} \times c^{11} d^{-6}
   \]
   Final answer:
   \[
   \frac{u^{14} s^{10} t^6 c^{11}}{9 d^6}
   \]

### Part (b):
\[
\left(\frac{e^{\frac{2}{3}}}{f^{-4}}\right)^{\frac{1}{2}} \times \left(9^{\frac{1}{2}} \times 16^{\frac{3}{2}}\right)
\]

1. **Simplify the first part:**
   Apply the exponent to both numerator and denominator:
   \[
   \left(\frac{e^{\frac{2}{3}}}{f^{-4}}\right)^{\frac{1}{2}} = \frac{e^{\frac{2}{3} \times \frac{1}{2}}}{f^{-4 \times \frac{1}{2}}} = \frac{e^{\frac{1}{3}}}{f^{-2}} = \frac{e^{\frac{1}{3}} f^2}{1}
   \]

2. **Now simplify the second part:**
   \[
   9^{\frac{1}{2}} = \sqrt{9} = 3, \quad 16^{\frac{3}{2}} = \sqrt{16^3} = \sqrt{4096} = 64
   \]
   So:
   \[
   9^{\frac{1}{2}} \times 16^{\frac{3}{2}} = 3 \times 64 = 192
   \]

3. **Combine both parts:**
   \[
   \frac{e^{\frac{1}{3}} f^2}{1} \times 192 = 192 e^{\frac{1}{3}} f^2
   \]

### Final Answers:
- (a) \(\frac{u^{14} s^{10} t^6 c^{11}}{9 d^6}\)
- (b) \(192 e^{\frac{1}{3}} f^2\)

Do you need further details or clarifications?

Here are 5 related questions:
1. How do you apply negative exponents to fractions?
2. What is the general rule for raising powers to powers?
3. Can you simplify fractional exponents to square or cube roots?
4. How do you manage variables with both negative and positive exponents?
5. How does changing the base affect the application of exponents?

**Tip:** Always handle negative exponents by flipping fractions before applying further operations, which simplifies the problem greatly.

Simplify each of the following, expressing the answers with positive indices.

Step-by-step guide to simplifying algebraic expressions involving fractions, negative exponents, and fractional exponents. Includes detailed solutions for simplifying terms with positive indices, suitable for Grades 9-11 students.

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