If `2^(x-3).3^(2x-8)=36`, then the value of x is ____________.

Options

A. 2
B. 5
C. 3
D. 1

Answer

To solve the equation \( 2^{(x-3)} \cdot 3^{(2x-8)} = 36 \), we will follow these steps:

### Step 1: Rewrite 36 in terms of its prime factors
We know that \( 36 = 6^2 = (2 \cdot 3)^2 = 2^2 \cdot 3^2 \).

### Step 2: Set up the equation
Now, we can rewrite the equation as:
\[ 2^{(x-3)} \cdot 3^{(2x-8)} = 2^2 \cdot 3^2 \]

### Step 3: Compare the exponents
Since the bases are the same, we can equate the exponents:
1. For the base 2:
\[ x - 3 = 2 \]
2. For the base 3:
\[ 2x - 8 = 2 \]

### Step 4: Solve the first equation
From \( x - 3 = 2 \):
\[ x = 2 + 3 = 5 \]

### Step 5: Solve the second equation
From \( 2x - 8 = 2 \):
\[ 2x = 2 + 8 \\ 2x = 10 \\ x = \frac{10}{2} = 5 \]

### Conclusion
Both equations give us the same value for \( x \). Therefore, the value of \( x \) is:
\[ \boxed{5} \]