The solution of `3^(3x - 5) = (1)/(9^x)` is

Options

A. `5/2`
B. 5
C. 1
D. `7/3`

Answer

To solve the equation \( 3^{3x - 5} = \frac{1}{9^x} \), we can follow these steps:

### Step 1: Rewrite \( 9^x \) in terms of base 3
We know that \( 9 = 3^2 \). Therefore, we can rewrite \( 9^x \) as:
\[ 9^x = (3^2)^x = 3^{2x} \]
So, we have:
\[ \frac{1}{9^x} = \frac{1}{3^{2x}} = 3^{-2x} \]

### Step 2: Rewrite the equation
Now, we can rewrite the original equation:
\[ 3^{3x - 5} = 3^{-2x} \]

### Step 3: Set the exponents equal to each other
Since the bases are the same (both are base 3), we can set the exponents equal to each other:
\[ 3x - 5 = -2x \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \) by isolating it:
1. Add \( 2x \) to both sides:
\[ 3x + 2x - 5 = 0 \implies 5x - 5 = 0 \]
2. Add 5 to both sides:
\[ 5x = 5 \]
3. Divide by 5:
\[ x = 1 \]

### Conclusion
The solution to the equation \( 3^{3x - 5} = \frac{1}{9^x} \) is:
\[ \boxed{1} \]

---