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!