Expressed with positive exponents only , `(ab^(-1))/(a^(-1) - b^(-1))` is equivalent to

Options

A. `(a^(2))/(a - b)`
B. `(a^(2)) /(a-1)`
C. `(b-a)/(ab)`
D. `(a^(2))/(b-a)`

Explanation

To solve the expression \(\frac{ab^{-1}}{a^{-1} - b^{-1}}\) and express it with positive exponents only, we can follow these steps:

### Step 1: Rewrite the expression using positive exponents
We start with the expression:
\[ \frac{ab^{-1}}{a^{-1} - b^{-1}} \]
We can rewrite \(b^{-1}\) as \(\frac{1}{b}\) and \(a^{-1}\) as \(\frac{1}{a}\):
\[ = \frac{a \cdot \frac{1}{b}}{\frac{1}{a} - \frac{1}{b}} \]

### Step 2: Simplify the denominator
The denominator \(\frac{1}{a} - \frac{1}{b}\) can be simplified by finding a common denominator:
\[ = \frac{b - a}{ab} \]
Thus, we can rewrite the expression as:
\[ = \frac{a \cdot \frac{1}{b}}{\frac{b - a}{ab}} = \frac{a}{b} \cdot \frac{ab}{b - a} \]

### Step 3: Cancel out terms
Now we can simplify the expression:
\[ = \frac{a \cdot ab}{b(b - a)} = \frac{a^2}{b - a} \]

### Step 4: Final expression
Now we have expressed the original expression with positive exponents only:
\[ = \frac{a^2}{b - a} \]

### Final Answer
Thus, the expression \(\frac{ab^{-1}}{a^{-1} - b^{-1}}\) is equivalent to:
\[ \frac{a^2}{b - a} \]