If `3^(x)-3^(x-1)=18`, then the value of `x^x` is

Options

A. `3`
B. `8`
C. `27`
D. `216`

Answer

To solve the equation \(3^x - 3^{x-1} = 18\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 3^x - 3^{x-1} = 18 \]
We can factor out \(3^{x-1}\):
\[ 3^{x-1}(3 - 1) = 18 \]

### Step 2: Simplify the equation
Now, simplify the expression:
\[ 3^{x-1} \cdot 2 = 18 \]
Dividing both sides by 2 gives:
\[ 3^{x-1} = \frac{18}{2} = 9 \]

### Step 3: Express 9 as a power of 3
We know that \(9\) can be expressed as a power of \(3\):
\[ 9 = 3^2 \]
Thus, we can rewrite the equation as:
\[ 3^{x-1} = 3^2 \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ x - 1 = 2 \]

### Step 5: Solve for \(x\)
Now, solve for \(x\):
\[ x = 2 + 1 = 3 \]

### Step 6: Calculate \(x^x\)
Now that we have \(x = 3\), we can find \(x^x\):
\[ x^x = 3^3 \]
Calculating \(3^3\):
\[ 3^3 = 3 \times 3 \times 3 = 27 \]

### Final Answer
Thus, the value of \(x^x\) is:
\[ \boxed{27} \]