Solve for x : `(a^(3x+3)xx(a^(3))^(4))=a^(8x+12)`

Answer

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

### Step 1: Simplify the left side of the equation
We can simplify the left side using the properties of exponents. The property states that \((a^m)^n = a^{m \cdot n}\).

\[ (a^3)^4 = a^{3 \cdot 4} = a^{12} \]

Thus, the left side becomes:

\[ a^{3x+3} \cdot a^{12} \]

### Step 2: Combine the exponents on the left side
Using the property \(a^m \cdot a^n = a^{m+n}\), we can combine the exponents:

\[ a^{(3x + 3) + 12} = a^{3x + 15} \]

Now, the equation looks like this:

\[ a^{3x + 15} = a^{8x + 12} \]

### Step 3: Set the exponents equal to each other
Since the bases are the same (both are \(a\)), we can set the exponents equal to each other:

\[ 3x + 15 = 8x + 12 \]

### Step 4: Solve for \(x\)
Now, we will solve the equation for \(x\). First, we can rearrange the equation:

\[ 3x + 15 - 12 = 8x \]

This simplifies to:

\[ 3x + 3 = 8x \]

Next, we can isolate \(x\) by moving \(3x\) to the right side:

\[ 3 = 8x - 3x \]

This simplifies to:

\[ 3 = 5x \]

Now, divide both sides by 5:

\[ x = \frac{3}{5} \]

### Final Answer
The solution for \(x\) is:

\[ x = \frac{3}{5} \]