If ` 2 ^(x +1) + 2 ^(x - 1) = 320,` then the value of x is :

Options

A. 8
B. 5
C. 7
D. 6

Answer

To solve the equation \( 2^{(x + 1)} + 2^{(x - 1)} = 320 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ 2^{(x + 1)} + 2^{(x - 1)} = 320 \]

### Step 2: Factor out the common term
Notice that we can factor out \( 2^{(x - 1)} \) from both terms:
\[ 2^{(x - 1)}(2^2 + 1) = 320 \]
This simplifies to:
\[ 2^{(x - 1)}(4 + 1) = 320 \]
Thus, we have:
\[ 2^{(x - 1)} \cdot 5 = 320 \]

### Step 3: Isolate \( 2^{(x - 1)} \)
Now, divide both sides by 5:
\[ 2^{(x - 1)} = \frac{320}{5} \]
Calculating the right side gives:
\[ 2^{(x - 1)} = 64 \]

### Step 4: Express 64 as a power of 2
We know that:
\[ 64 = 2^6 \]
So we can set the exponents equal to each other:
\[ x - 1 = 6 \]

### Step 5: Solve for \( x \)
Now, add 1 to both sides:
\[ x = 6 + 1 \]
Thus, we find:
\[ x = 7 \]

### Final Answer
The value of \( x \) is \( 7 \).

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