If `2^(x+4)-2^(x+2)=3`, then x is equal to

Options

A. 0
B. 2
C. `-1`
D. `-2`

Answer

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

### Step 1: Rewrite the equation
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 \( 2^{(x+2)} \):
\[ 2^{(x+2)}(2^2 - 1) = 3 \]
Here, \( 2^2 = 4 \), so we can simplify it further:
\[ 2^{(x+2)}(4 - 1) = 3 \]
This simplifies to:
\[ 2^{(x+2)} \cdot 3 = 3 \]

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

### Step 4: Rewrite 1 as a power of 2
We know that \( 1 \) can be expressed as \( 2^0 \):
\[ 2^{(x+2)} = 2^0 \]

### Step 5: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ x + 2 = 0 \]

### Step 6: Solve for x
Now, solve for \( x \):
\[ x = 0 - 2 \]
\[ x = -2 \]

Thus, the value of \( x \) is \( -2 \).

### Final Answer:
\[ x = -2 \]

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