Solve for x :
`3^(4x + 1) = (27)^(x + 1)`

Answer

To solve the equation \(3^{4x + 1} = (27)^{x + 1}\), we can follow these steps:

### Step 1: Rewrite the base on the right side
We know that \(27\) can be expressed as a power of \(3\):
\[ 27 = 3^3 \]
Thus, we can rewrite the equation as:
\[ 3^{4x + 1} = (3^3)^{x + 1} \]

### Step 2: Apply the power of a power property
Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\), we can simplify the right side:
\[ (3^3)^{x + 1} = 3^{3(x + 1)} \]
So the equation now looks like:
\[ 3^{4x + 1} = 3^{3(x + 1)} \]

### Step 3: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 4x + 1 = 3(x + 1) \]

### Step 4: Expand and simplify the equation
Now, we will expand the right side:
\[ 4x + 1 = 3x + 3 \]
Next, we will isolate \(x\) by moving terms involving \(x\) to one side and constant terms to the other:
\[ 4x - 3x = 3 - 1 \]
This simplifies to:
\[ x = 2 \]

### Final Answer
Thus, the solution for \(x\) is:
\[ \boxed{2} \]