If `((2^(n+4)-2.2^(n))/(2.2^(n+3) )+ 2^(-3) )=x`, then the value of x is

Options

A. a. `-2^(n+1)+1/8`
B. b. 1
C. c. `2^(n+1)`
D. d. `n/8-2^(n)`

Answer

To solve the expression given in the question, we will follow these steps:

### Step 1: Write the expression clearly
We start with the expression:
\[ \frac{2^{n+4} - 2 \cdot 2^n}{2 \cdot 2^{n+3}} + 2^{-3} = x \]

### Step 2: Simplify the numerator
In the numerator \(2^{n+4} - 2 \cdot 2^n\), we can factor out \(2^n\):
\[ 2^{n+4} - 2 \cdot 2^n = 2^n(2^4 - 2) = 2^n(16 - 2) = 2^n \cdot 14 \]

### Step 3: Simplify the denominator
In the denominator \(2 \cdot 2^{n+3}\), we can simplify it as:
\[ 2 \cdot 2^{n+3} = 2^{1} \cdot 2^{n+3} = 2^{n+4} \]

### Step 4: Rewrite the expression
Now we can rewrite the expression as:
\[ \frac{2^n \cdot 14}{2^{n+4}} + 2^{-3} = x \]

### Step 5: Simplify the fraction
The fraction simplifies to:
\[ \frac{14}{2^{4}} = \frac{14}{16} = \frac{7}{8} \]

### Step 6: Add \(2^{-3}\)
Next, we need to add \(2^{-3}\) to \(\frac{7}{8}\):
\[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]

Now we can add:
\[ \frac{7}{8} + \frac{1}{8} = \frac{7 + 1}{8} = \frac{8}{8} = 1 \]

### Step 7: Conclusion
Thus, we find that:
\[ x = 1 \]

### Final Answer
The value of \(x\) is:
\[ \boxed{1} \]

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