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Let's simplify the given expression step by step. The expression is:

\[
\frac{4^{n+3} \times 2^{4-n}}{8^{n+1}}
\]

### Step 1: Express all terms with the same base
First, we convert each term to a power of 2:
- \( 4^{n+3} = (2^2)^{n+3} = 2^{2(n+3)} = 2^{2n+6} \)
- \( 8^{n+1} = (2^3)^{n+1} = 2^{3(n+1)} = 2^{3n+3} \)

So, the expression becomes:
\[
\frac{2^{2n+6} \times 2^{4-n}}{2^{3n+3}}
\]

### Step 2: Simplify the multiplication in the numerator
Combine the exponents in the numerator using the property \( a^m \times a^n = a^{m+n} \):
\[
2^{(2n+6) + (4-n)} = 2^{2n+6+4-n} = 2^{n+10}
\]

### Step 3: Simplify the division
Now, simplify the expression using the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{2^{n+10}}{2^{3n+3}} = 2^{(n+10) - (3n+3)} = 2^{n+10-3n-3} = 2^{-2n+7}
\]

So, the simplified expression is:
\[
2^{7-2n}
\]

### Final Answer
The simplified form of the given expression is:

\[
2^{7-2n}
\]

Would you like further details or have any questions?

Here are some related questions for practice:
1. How would you simplify \(\frac{3^{2n+1} \times 9^{n-2}}{27^n}\)?
2. What is the general process for simplifying expressions with different bases?
3. Can you convert \(5^{3n} \times 25^{n+1}\) to a single power of 5?
4. How would you approach simplifying \(2^{4n-2} \times 8^{n+1}\)?
5. How does the property \( \frac{a^m}{a^n} = a^{m-n} \) help in simplifying expressions?

**Tip:** Always look to express terms with a common base before simplifying. It makes the process much more straightforward!

Simplify the expression: (4^(n+3) * 2^(4-n)) / (8^(n+1))

Learn how to simplify the expression (4^(n+3) * 2^(4-n)) / (8^(n+1)) using exponent laws. This step-by-step guide covers key algebraic concepts and is perfect for students in Grades 9-10.

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