Solve for x :
`2^((3x + 3)) = 2^((3x + 1)) + 48`.

Answer

To solve the equation \( 2^{(3x + 3)} = 2^{(3x + 1)} + 48 \), we can follow these steps:

### Step 1: Rewrite the equation using exponent rules
We can use the property of exponents that states \( a^{(b+c)} = a^b \cdot a^c \). Thus, we can rewrite the left side of the equation:
\[ 2^{(3x + 3)} = 2^{(3x)} \cdot 2^3 \]
This simplifies to:
\[ 2^{(3x + 3)} = 2^{(3x)} \cdot 8 \]

### Step 2: Rewrite the right side of the equation
Similarly, we can rewrite the right side:
\[ 2^{(3x + 1)} = 2^{(3x)} \cdot 2^1 \]
This simplifies to:
\[ 2^{(3x + 1)} = 2^{(3x)} \cdot 2 \]

### Step 3: Substitute the rewritten forms into the equation
Now we substitute these forms back into the original equation:
\[ 2^{(3x)} \cdot 8 = 2^{(3x)} \cdot 2 + 48 \]

### Step 4: Let \( k = 2^{(3x)} \)
To simplify the equation further, let \( k = 2^{(3x)} \). The equation now becomes:
\[ 8k = 2k + 48 \]

### Step 5: Rearrange the equation
Now, we can rearrange the equation:
\[ 8k - 2k = 48 \]
This simplifies to:
\[ 6k = 48 \]

### Step 6: Solve for \( k \)
Now, divide both sides by 6:
\[ k = \frac{48}{6} = 8 \]

### Step 7: Substitute back for \( k \)
Recall that \( k = 2^{(3x)} \). So we have:
\[ 2^{(3x)} = 8 \]

### Step 8: Rewrite 8 as a power of 2
Since \( 8 = 2^3 \), we can write:
\[ 2^{(3x)} = 2^3 \]

### Step 9: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 3x = 3 \]

### Step 10: Solve for \( x \)
Now, divide both sides by 3:
\[ x = 1 \]

### Final Answer
Thus, the solution for \( x \) is:
\[ \boxed{1} \]