If `(ab^(-1))^(2x-1)=(ba^(-1))^(x-2)` then what is the value of x?

Options

A. 1
B. 2
C. 3
D. 4

Answer

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ (ab^{-1})^{2x-1} = (ba^{-1})^{x-2} \]

### Step 2: Express negative exponents as fractions
Using the property of negative exponents, we can rewrite \(b^{-1}\) and \(a^{-1}\):
\[ (ab^{-1})^{2x-1} = \left(\frac{a}{b}\right)^{2x-1} \quad \text{and} \quad (ba^{-1})^{x-2} = \left(\frac{b}{a}\right)^{x-2} \]
Thus, we can rewrite the equation as:
\[ \left(\frac{a}{b}\right)^{2x-1} = \left(\frac{b}{a}\right)^{x-2} \]

### Step 3: Rewrite the right side with a negative exponent
The right side can be rewritten using the property of reciprocals:
\[ \left(\frac{b}{a}\right)^{x-2} = \left(\frac{a}{b}\right)^{-(x-2)} \]
So now we have:
\[ \left(\frac{a}{b}\right)^{2x-1} = \left(\frac{a}{b}\right)^{-(x-2)} \]

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

### Step 5: Simplify the equation
Now we simplify the equation:
\[ 2x - 1 = -x + 2 \]
Adding \(x\) to both sides:
\[ 2x + x - 1 = 2 \]
This simplifies to:
\[ 3x - 1 = 2 \]
Adding \(1\) to both sides:
\[ 3x = 3 \]
Dividing both sides by \(3\):
\[ x = 1 \]

### Final Answer
The value of \(x\) is:
\[ \boxed{1} \]