If `2^(x-1)+2^(x+1)=2560`, find the value of x.

Options

A. 10
B. 12
C. 9
D. 8

Answer

To solve the equation \(2^{(x-1)} + 2^{(x+1)} = 2560\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 2^{(x-1)} + 2^{(x+1)} = 2560 \]

### Step 2: Factor out the common term
Notice that \(2^{(x+1)}\) can be rewritten as \(2^{(x-1)} \cdot 2^2\). Thus, we can factor out \(2^{(x-1)}\):
\[ 2^{(x-1)} + 2^{(x-1)} \cdot 2^2 = 2560 \]
This simplifies to:
\[ 2^{(x-1)}(1 + 4) = 2560 \]
\[ 2^{(x-1)} \cdot 5 = 2560 \]

### Step 3: Isolate \(2^{(x-1)}\)
Now, we can isolate \(2^{(x-1)}\):
\[ 2^{(x-1)} = \frac{2560}{5} \]
Calculating the right side:
\[ 2^{(x-1)} = 512 \]

### Step 4: Rewrite \(512\) as a power of \(2\)
We know that \(512\) can be expressed as a power of \(2\):
\[ 512 = 2^9 \]
Thus, we have:
\[ 2^{(x-1)} = 2^9 \]

### Step 5: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ x - 1 = 9 \]

### Step 6: Solve for \(x\)
Now, we solve for \(x\):
\[ x = 9 + 1 = 10 \]

### Final Answer
The value of \(x\) is:
\[ \boxed{10} \]

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