Solve for x :
`(a^(3x+5))^(2) . (a^(x))^(4) = a^(8x+12).`

Answer

To solve the equation \((a^{3x+5})^{2} \cdot (a^{x})^{4} = a^{8x+12}\), we will follow these steps:

### Step 1: Apply the Power of a Power Rule
Using the identity \((a^m)^n = a^{m \cdot n}\), we can simplify the left side of the equation:

\[ (a^{3x+5})^{2} = a^{2(3x+5)} = a^{6x + 10} \]

\[ (a^{x})^{4} = a^{4x} \]

### Step 2: Combine the Exponents
Now, we can rewrite the left side of the equation by combining the exponents:

\[ a^{6x + 10} \cdot a^{4x} = a^{(6x + 10) + 4x} = a^{10x + 10} \]

### Step 3: Set the Exponents Equal
Now that we have simplified both sides, we can set the exponents equal to each other since the bases are the same:

\[ 10x + 10 = 8x + 12 \]

### Step 4: Solve for \(x\)
Now, we will solve for \(x\):

1. Subtract \(8x\) from both sides:
\[ 10x - 8x + 10 = 12 \]
\[ 2x + 10 = 12 \]

2. Subtract \(10\) from both sides:
\[ 2x = 12 - 10 \]
\[ 2x = 2 \]

3. Divide both sides by \(2\):
\[ x = 1 \]

### Final Answer
Thus, the solution for \(x\) is:

\[ \boxed{1} \]