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The expression to simplify is:

\[
\frac{2^{x+5} \cdot 16^{x+4}}{8^{x+3} \cdot 4^{x+2}}
\]

Let's simplify this step by step.

### Step 1: Rewrite the bases in terms of powers of 2

- \( 2^{x+5} \) is already in base 2.
- \( 16 = 2^4 \), so \( 16^{x+4} = (2^4)^{x+4} = 2^{4(x+4)} = 2^{4x + 16} \).
- \( 8 = 2^3 \), so \( 8^{x+3} = (2^3)^{x+3} = 2^{3(x+3)} = 2^{3x + 9} \).
- \( 4 = 2^2 \), so \( 4^{x+2} = (2^2)^{x+2} = 2^{2(x+2)} = 2^{2x + 4} \).

### Step 2: Substitute back into the expression

Now we have:

\[
\frac{2^{x+5} \cdot 2^{4x + 16}}{2^{3x + 9} \cdot 2^{2x + 4}}
\]

### Step 3: Simplify the exponents

Combine the exponents in the numerator and denominator:

- Numerator: \( 2^{x+5} \cdot 2^{4x+16} = 2^{(x+5) + (4x+16)} = 2^{5x + 21} \).
- Denominator: \( 2^{3x+9} \cdot 2^{2x+4} = 2^{(3x+9) + (2x+4)} = 2^{5x + 13} \).

### Step 4: Simplify the overall expression

Now the expression becomes:

\[
\frac{2^{5x + 21}}{2^{5x + 13}} = 2^{(5x + 21) - (5x + 13)} = 2^{8}
\]

### Final Answer

The simplified expression is:

\[
2^8 = 256
\]

Would you like any more details or have any questions?

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Here are five related questions for further exploration:

1. How would the simplification process change if the bases were not powers of 2?
2. What happens if you multiply or divide the entire expression by another term?
3. Can you generalize the steps for simplifying expressions with other bases?
4. How does understanding exponents help in solving logarithmic equations?
5. What is the significance of the exponent rules in different areas of mathematics?

**Tip:** When dealing with expressions involving exponents, always try to express all terms with the same base—it greatly simplifies the process.

Explore how to simplify the expression (2^(x+5) * 16^(x+4)) / (8^(x+3) * 4^(x+2)) using detailed steps. Learn about exponents, powers of 2, and simplification techniques in algebra. Suitable for high school students.

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