The value of x for which `2^(x+4)-2^(x+2)=3` is

Options

A. `-2`
B. `-3`
C. 0
D. 5

Answer

To solve the equation \( 2^{(x+4)} - 2^{(x+2)} = 3 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ 2^{(x+4)} - 2^{(x+2)} = 3 \]

### Step 2: Factor out the common term
Notice that both terms on the left-hand side have a common factor of \( 2^{(x+2)} \):
\[ 2^{(x+2)}(2^2 - 1) = 3 \]
This simplifies to:
\[ 2^{(x+2)}(4 - 1) = 3 \]
\[ 2^{(x+2)}(3) = 3 \]

### Step 3: Simplify the equation
Now, we can divide both sides by 3:
\[ 2^{(x+2)} = 1 \]

### Step 4: Solve for \( x \)
Since \( 2^{(x+2)} = 1 \), we know that:
\[ 2^0 = 1 \]
This implies:
\[ x + 2 = 0 \]
Thus, solving for \( x \):
\[ x = -2 \]

### Final Answer
The value of \( x \) is:
\[ \boxed{-2} \]