(16 * 2 ^ (n + 1) - 4 * 2 ^ n)/(16 * 2 ^ (n + 1) - 2 * 2 ^ (n + 2)) =

(a) - 1/4

(b) - 1/2

1/2 (0)

(P) 1/4

To simplify the given expression:

\[
\frac{16 \cdot 2^{n+1} - 4 \cdot 2^n}{16 \cdot 2^{n+1} - 2 \cdot 2^{n+2}}
\]

we can start by simplifying both the numerator and the denominator.

### Step 1: Simplify the numerator

The numerator is:

\[
16 \cdot 2^{n+1} - 4 \cdot 2^n
\]

Factor out the common term \(2^n\):

\[
= 2^n (16 \cdot 2 - 4)
\]

Calculate the terms inside the parentheses:

\[
16 \cdot 2 = 32, \quad 32 - 4 = 28
\]

So, the simplified numerator is:

\[
2^n \cdot 28 = 28 \cdot 2^n
\]

### Step 2: Simplify the denominator

The denominator is:

\[
16 \cdot 2^{n+1} - 2 \cdot 2^{n+2}
\]

Rewrite \(2^{n+2}\) as \(4 \cdot 2^n\) and factor out the common term \(2^n\):

\[
= 2^n (16 \cdot 2 - 2 \cdot 4)
\]

Calculate the terms inside the parentheses:

\[
16 \cdot 2 = 32, \quad 2 \cdot 4 = 8, \quad 32 - 8 = 24
\]

So, the simplified denominator is:

\[
2^n \cdot 24 = 24 \cdot 2^n
\]

### Step 3: Simplify the fraction

Now the expression is:

\[
\frac{28 \cdot 2^n}{24 \cdot 2^n}
\]

Since \(2^n\) appears in both the numerator and the denominator, we can cancel it:

\[
\frac{28}{24}
\]

Simplify the fraction by dividing both the numerator and the denominator by 4:

\[
\frac{28 \div 4}{24 \div 4} = \frac{7}{6}
\]

### Conclusion

The simplified value of the expression is \(\frac{7}{6}\), which does not match any of the provided options exactly. This indicates a possible error in the options given or a misunderstanding of the problem context.

If the answer choices or context were incorrect, here’s the correct simplified form:

\[
\frac{7}{6}
\]

Do you have any more details or need further clarifications?

### Relative Questions:

1. What if the original expression had different coefficients?
2. How would you simplify a fraction involving higher powers of 2?
3. Can this problem be interpreted in terms of functions?
4. What would happen if \(n\) were a specific number, like \(n = 2\)?
5. Are there other methods to simplify expressions with exponential terms?

**Tip:** Always double-check the expression for any possible errors or misinterpretations when the provided answer choices don’t match.

Learn how to simplify the exponential expression (16 * 2^(n + 1) - 4 * 2^n) / (16 * 2^(n + 1) - 2 * 2^(n + 2)) step-by-step. This problem involves exponential functions and simplifying complex fractions. Suitable for advanced high school students studying algebra and exponential functions.

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