Solve the following expressions .
`2^(2x-1) = 1/(8^(x - 3))`, then x = ?

Answer

To solve the equation \( 2^{(2x-1)} = \frac{1}{8^{(x - 3)}} \), we will follow these steps:

### Step 1: Rewrite the Right Side
We can rewrite \( \frac{1}{8^{(x - 3)}} \) using the property of exponents that states \( \frac{1}{a^b} = a^{-b} \). Thus, we have:
\[ \frac{1}{8^{(x - 3)}} = 8^{-(x - 3)} \]

### Step 2: Express 8 in Terms of Base 2
Next, we express \( 8 \) as \( 2^3 \):
\[ 8^{-(x - 3)} = (2^3)^{-(x - 3)} = 2^{-3(x - 3)} \]
This simplifies to:
\[ 2^{-3x + 9} \]

### Step 3: Set the Equation
Now, we can set the left side equal to the right side:
\[ 2^{(2x-1)} = 2^{-3x + 9} \]

### Step 4: Equate the Exponents
Since the bases are the same, we can equate the exponents:
\[ 2x - 1 = -3x + 9 \]

### Step 5: Solve for x
Now, let's solve for \( x \):
1. Add \( 3x \) to both sides:
\[ 2x + 3x - 1 = 9 \]
This simplifies to:
\[ 5x - 1 = 9 \]
2. Add \( 1 \) to both sides:
\[ 5x = 10 \]
3. Divide by \( 5 \):
\[ x = 2 \]

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{2} \]