Let's analyze the given mathematical expression:

$\frac{2^4 \cdot 2^3 \cdot 2^{-3}}{\sqrt{4^2 \cdot 2^4 \cdot 2^{-3}}} = 2^n$

We need to simplify this expression step by step to find the value of $n$.

Step 1: Simplifying the numerator

The numerator is:

$2^4 \cdot 2^3 \cdot 2^{-3}$

Using the property of exponents $a^m \cdot a^n = a^{m+n}$, we combine the powers of 2:

$2^4 \cdot 2^3 \cdot 2^{-3} = 2^{4 + 3 - 3} = 2^4$

Step 2: Simplifying the denominator

The denominator is the square root expression:

$\sqrt{4^2 \cdot 2^4 \cdot 2^{-3}}$

First, express $4^2$ as $(2^2)^2 = 2^4$:

$\sqrt{2^4 \cdot 2^4 \cdot 2^{-3}} = \sqrt{2^{4+4-3}} = \sqrt{2^5}$

Now, simplify the square root of $2^5$:

$\sqrt{2^5} = 2^{5/2}$

Step 3: Putting it all together

Now the expression becomes:

$\frac{2^4}{2^{5/2}}$

Using the property $\frac{a^m}{a^n} = a^{m-n}$:

$2^{4 - 5/2} = 2^{\frac{8}{2} - \frac{5}{2}} = 2^{\frac{3}{2}}$

Thus, we have:

$2^{n} = 2^{\frac{3}{2}}$

Step 4: Conclusion

Therefore, $n = \frac{3}{2}$.

Would you like a detailed explanation of any specific step or have further questions?

Here are 5 related questions for practice:

1. How do you simplify the product of exponents with the same base?
2. What is the square root of an expression involving exponents?
3. How would you approach a similar problem if the base was different from 2?
4. Can you simplify $\frac{a^m}{a^n}$ for general values of $m$ and $n$?
5. How does the fractional exponent $a^{1/n}$ relate to roots?

Tip: When simplifying expressions with exponents, always combine terms with the same base using the properties of exponents.