If `2 ^(x +1) + 2 ^( x-1)= 80,` then find the value of x.

Options

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

Answer

To solve the equation \(2^{(x + 1)} + 2^{(x - 1)} = 80\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 2^{(x + 1)} + 2^{(x - 1)} = 80 \]
We can rewrite \(2^{(x + 1)}\) as \(2^x \cdot 2^1 = 2 \cdot 2^x\) and \(2^{(x - 1)}\) as \(2^x \cdot 2^{-1} = \frac{2^x}{2}\).

So, the equation becomes:
\[ 2 \cdot 2^x + \frac{2^x}{2} = 80 \]

### Step 2: Factor out \(2^x\)
Now, we can factor out \(2^x\):
\[ 2^x \left(2 + \frac{1}{2}\right) = 80 \]

### Step 3: Simplify the expression in the parentheses
Next, simplify the expression in the parentheses:
\[ 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \]
Thus, we have:
\[ 2^x \cdot \frac{5}{2} = 80 \]

### Step 4: Isolate \(2^x\)
To isolate \(2^x\), multiply both sides by \(\frac{2}{5}\):
\[ 2^x = 80 \cdot \frac{2}{5} \]
Calculating the right side:
\[ 80 \cdot \frac{2}{5} = 16 \cdot 2 = 32 \]
So, we have:
\[ 2^x = 32 \]

### Step 5: Solve for \(x\)
Since \(32\) can be expressed as a power of \(2\):
\[ 32 = 2^5 \]
Thus, we equate the exponents:
\[ x = 5 \]

### Final Answer
The value of \(x\) is:
\[ \boxed{5} \]