To solve the given expression:

\[
8^{2f+1} \times 16^{4-f} \div 4^{f+2}
\]

we need to simplify it step-by-step. Let's rewrite each term using a common base.

### Step 1: Rewrite Terms with Base 2

- \(8 = 2^3\), so \(8^{2f+1} = (2^3)^{2f+1} = 2^{3(2f+1)} = 2^{6f+3}\)
- \(16 = 2^4\), so \(16^{4-f} = (2^4)^{4-f} = 2^{4(4-f)} = 2^{16-4f}\)
- \(4 = 2^2\), so \(4^{f+2} = (2^2)^{f+2} = 2^{2(f+2)} = 2^{2f+4}\)

### Step 2: Rewrite the Expression with Base 2

\[
2^{6f+3} \times 2^{16-4f} \div 2^{2f+4}
\]

### Step 3: Apply the Laws of Exponents

Using the laws of exponents:

- Multiplication: \(a^m \times a^n = a^{m+n}\)
- Division: \(\frac{a^m}{a^n} = a^{m-n}\)

We can simplify the expression:

\[
2^{(6f+3) + (16-4f)} \div 2^{2f+4}
\]

Combine the exponents in the numerator:

\[
2^{6f + 3 + 16 - 4f} = 2^{2f + 19}
\]

Now, divide by \(2^{2f+4}\):

\[
2^{(2f + 19) - (2f + 4)} = 2^{2f + 19 - 2f - 4} = 2^{15}
\]

### Conclusion

The simplified form of the expression is:

\[
2^{15}
\]

Or, in decimal form:

\[
2^{15} = 32,768
\]

### Final Answer

\[
\boxed{32,768}
\]

Would you like more details or have any questions? 

Here are 5 related questions to explore further:

1. How can we simplify expressions with multiple exponents of different bases?
2. What are the rules for multiplying and dividing exponents?
3. How can we convert between different bases, like base 2 to base 10?
4. What is the significance of exponent rules in algebra?
5. How can we solve equations involving exponential growth and decay?

**Tip:** When working with exponents, always look for a common base to simplify the calculations.

Learn how to simplify the complex exponential expression 8^(2f+1) x 16^(4-f) / 4^(f+2) step-by-step. This problem covers key algebraic concepts like exponent rules and simplification techniques. Suitable for students in Grades 10-12.

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