Which of the following is equivalent of `((ab)/(c))(cb-a)`?

Options

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

Answer

To solve the expression \(\frac{ab}{c}(cb - a)\), we will follow these steps:

### Step 1: Distribute the fraction
We start by distributing \(\frac{ab}{c}\) to both terms inside the parentheses:

\[ \frac{ab}{c}(cb - a) = \frac{ab}{c} \cdot cb - \frac{ab}{c} \cdot a \]

### Step 2: Simplify the first term
Now, we simplify the first term:

\[ \frac{ab \cdot cb}{c} = \frac{a \cdot b \cdot c \cdot b}{c} \]

Here, the \(c\) in the numerator and denominator cancels out:

\[ = ab^2 \]

### Step 3: Simplify the second term
Next, we simplify the second term:

\[ \frac{ab \cdot a}{c} = \frac{a^2b}{c} \]

### Step 4: Combine the terms
Now we combine both simplified terms:

\[ ab^2 - \frac{a^2b}{c} \]

### Step 5: Write the final expression
Thus, the expression simplifies to:

\[ \frac{ab^2 - a^2b}{c} \]

### Conclusion
The equivalent expression of \(\frac{ab}{c}(cb - a)\) is:

\[ \frac{ab^2 - a^2b}{c} \]