If `3^(x)-3^(x-1)=18`, then `x^(x)` is equal to

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 can rewrite the term \( 3^{x-1} \) as \( \frac{3^x}{3} \). Thus, the equation becomes:
\[ 3^x - \frac{3^x}{3} = 18 \]

### Step 2: Simplify the equation
Now, factor out \( 3^x \):
\[ 3^x \left(1 - \frac{1}{3}\right) = 18 \]
This simplifies to:
\[ 3^x \left(\frac{2}{3}\right) = 18 \]

### Step 3: Multiply both sides by \( \frac{3}{2} \)
To isolate \( 3^x \), we multiply both sides by \( \frac{3}{2} \):
\[ 3^x = 18 \cdot \frac{3}{2} \]
Calculating the right side:
\[ 3^x = 27 \]

### Step 4: Rewrite \( 27 \) as a power of \( 3 \)
We know that \( 27 = 3^3 \). Therefore, we can write:
\[ 3^x = 3^3 \]

### Step 5: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ x = 3 \]

### Step 6: Calculate \( x^x \)
Now, we need to find \( x^x \):
\[ x^x = 3^3 \]
Calculating \( 3^3 \):
\[ 3^3 = 27 \]

### Final Answer
Thus, the value of \( x^x \) is \( 27 \).

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