https://file.aiquickdraw.com/mathbot/file/1725296329134mjm3pr61.jpg

Let's simplify each of the expressions from parts (d), (e), and (f).

### (d) \(2^{x-1} \times 4^{3x-2} \div 32^{x-1}\)

1. **Rewrite the bases in terms of 2:**
   \[
   4 = 2^2 \quad \text{and} \quad 32 = 2^5
   \]
   So, rewrite the expression:
   \[
   2^{x-1} \times (2^2)^{3x-2} \div (2^5)^{x-1}
   \]

2. **Simplify the exponents:**
   \[
   2^{x-1} \times 2^{2(3x-2)} \div 2^{5(x-1)}
   \]
   Simplify the exponents:
   \[
   2^{x-1} \times 2^{6x-4} \div 2^{5x-5}
   \]

3. **Combine exponents using properties of exponents:**
   \[
   2^{(x-1) + (6x-4) - (5x-5)}
   \]
   Simplify the exponent:
   \[
   x - 1 + 6x - 4 - 5x + 5 = 2x
   \]

4. **Final simplified form:**
   \[
   2^{2x}
   \]

### (e) \(6^x \times 12^{x-2} \div 27^x \times 32^x\)

1. **Rewrite the bases in terms of their prime factors:**
   \[
   6 = 2 \cdot 3, \quad 12 = 2^2 \cdot 3, \quad 27 = 3^3, \quad 32 = 2^5
   \]
   So, rewrite the expression:
   \[
   (2 \cdot 3)^x \times (2^2 \cdot 3)^{x-2} \div (3^3)^x \times (2^5)^x
   \]

2. **Expand and simplify the exponents:**
   \[
   2^x \cdot 3^x \times 2^{2(x-2)} \cdot 3^{x-2} \div 3^{3x} \times 2^{5x}
   \]

   Combine the powers of 2 and 3:
   \[
   \frac{2^x \cdot 2^{2(x-2)} \cdot 3^x \cdot 3^{x-2}}{3^{3x} \cdot 2^{5x}}
   \]

   Simplify each base separately:
   \[
   \frac{2^{x + 2(x-2)} \cdot 3^{x + (x-2)}}{2^{5x} \cdot 3^{3x}}
   \]

   Simplify exponents:
   \[
   \frac{2^{x + 2x - 4} \cdot 3^{x + x - 2}}{2^{5x} \cdot 3^{3x}}
   \]
   \[
   \frac{2^{3x - 4} \cdot 3^{2x - 2}}{2^{5x} \cdot 3^{3x}}
   \]

3. **Combine exponents using properties of exponents:**
   \[
   2^{(3x - 4 - 5x)} \cdot 3^{(2x - 2 - 3x)}
   \]

   Simplify the exponents:
   \[
   2^{-2x - 4} \cdot 3^{-x - 2}
   \]

4. **Final simplified form:**
   \[
   \frac{1}{2^{2x + 4} \cdot 3^{x + 2}}
   \]

### (f) \(20^{x+3} \times 15^{2x+5} \div 6^{2x-1}\)

1. **Rewrite the bases in terms of their prime factors:**
   \[
   20 = 2^2 \cdot 5, \quad 15 = 3 \cdot 5, \quad 6 = 2 \cdot 3
   \]
   So, rewrite the expression:
   \[
   (2^2 \cdot 5)^{x+3} \times (3 \cdot 5)^{2x+5} \div (2 \cdot 3)^{2x-1}
   \]

2. **Expand and simplify the exponents:**
   \[
   2^{2(x+3)} \cdot 5^{x+3} \times 3^{2x+5} \cdot 5^{2x+5} \div (2^{2x-1} \cdot 3^{2x-1})
   \]

   Combine the powers of 2, 3, and 5:
   \[
   \frac{2^{2(x+3)} \cdot 5^{x+3 + 2x+5} \cdot 3^{2x+5}}{2^{2x-1} \cdot 3^{2x-1}}
   \]

   Simplify exponents:
   \[
   \frac{2^{2x + 6} \cdot 5^{3x + 8} \cdot 3^{2x + 5}}{2^{2x - 1} \cdot 3^{2x - 1}}
   \]

3. **Combine exponents using properties of exponents:**
   \[
   2^{(2x + 6 - 2x + 1)} \cdot 3^{(2x + 5 - 2x + 1)} \cdot 5^{3x + 8}
   \]

   Simplify the exponents:
   \[
   2^7 \cdot 3^6 \cdot 5^{3x+8}
   \]

4. **Final simplified form:**
   \[
   128 \cdot 729 \cdot 5^{3x+8}
   \]
   or more cleanly:
   \[
   93312 \cdot 5^{3x+8}
   \]

If you have any questions or would like more details on any of these steps, please let me know!

### Related Questions:

1. What is the importance of understanding the properties of exponents in simplifying expressions?
2. How can you convert expressions involving roots into exponents for simplification?
3. What are some common mistakes to avoid when applying exponent rules?
4. How can these simplifications be applied in solving algebraic equations?
5. How do these properties of exponents relate to logarithmic functions?

**Tip:** Always rewrite expressions with the same base to simplify them effectively using exponent rules.

Explore how to simplify exponential expressions involving various bases and exponents. This detailed solution covers problems (d), (e), and (f) with step-by-step explanations, focusing on exponent rules and prime factorization. Suitable for high school students learning advanced algebra concepts.

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