In the equation `4^(x + 2) = 2^(x +3) + 48` , the value of x will be

Options

A. -3/2
B. -2
C. -3
D. 1

Answer

To solve the equation \( 4^{(x + 2)} = 2^{(x + 3)} + 48 \), we can follow these steps:

### Step 1: Rewrite the equation in terms of base 2
Since \( 4 \) can be expressed as \( 2^2 \), we can rewrite the left side of the equation:
\[ 4^{(x + 2)} = (2^2)^{(x + 2)} = 2^{2(x + 2)} = 2^{(2x + 4)} \]
Thus, the equation becomes:
\[ 2^{(2x + 4)} = 2^{(x + 3)} + 48 \]

### Step 2: Set up the equation
Now we have:
\[ 2^{(2x + 4)} = 2^{(x + 3)} + 48 \]

### Step 3: Test possible integer values for \( x \)
We will test integer values for \( x \) to find a solution.

1. **Testing \( x = -2 \)**:
\[ 2^{(2(-2) + 4)} = 2^{(0)} = 1 \]
\[ 2^{(-2 + 3)} + 48 = 2^{(1)} + 48 = 2 + 48 = 50 \]
Not equal.

2. **Testing \( x = -1 \)**:
\[ 2^{(2(-1) + 4)} = 2^{(2)} = 4 \]
\[ 2^{(-1 + 3)} + 48 = 2^{(2)} + 48 = 4 + 48 = 52 \]
Not equal.

3. **Testing \( x = 0 \)**:
\[ 2^{(2(0) + 4)} = 2^{(4)} = 16 \]
\[ 2^{(0 + 3)} + 48 = 2^{(3)} + 48 = 8 + 48 = 56 \]
Not equal.

4. **Testing \( x = 1 \)**:
\[ 2^{(2(1) + 4)} = 2^{(6)} = 64 \]
\[ 2^{(1 + 3)} + 48 = 2^{(4)} + 48 = 16 + 48 = 64 \]
Equal! So, \( x = 1 \) is a solution.

### Conclusion
The value of \( x \) that satisfies the equation \( 4^{(x + 2)} = 2^{(x + 3)} + 48 \) is:
\[ \boxed{1} \]