Simplify : `(12ab^(3)-6a^(2)b)/(3ab)` (given that `ab ne 0`)

Answer

To simplify the expression \((12ab^3 - 6a^2b)/(3ab)\), we can follow these steps:

### Step 1: Split the Fraction
We can separate the terms in the numerator over the common denominator:
\[ \frac{12ab^3 - 6a^2b}{3ab} = \frac{12ab^3}{3ab} - \frac{6a^2b}{3ab} \]

### Step 2: Simplify Each Term
Now, we simplify each term separately.

**For the first term:**
\[ \frac{12ab^3}{3ab} \]
- The \(a\) in the numerator and denominator cancels out.
- The \(b\) in the numerator \(b^3\) and the \(b\) in the denominator cancels to give \(b^{3-1} = b^2\).
- The numerical part simplifies as \(12/3 = 4\).

Thus, the first term simplifies to:
\[ 4b^2 \]

**For the second term:**
\[ \frac{6a^2b}{3ab} \]
- Again, the \(a\) cancels out.
- The \(b\) in the numerator cancels with the \(b\) in the denominator to give \(1\).
- The numerical part simplifies as \(6/3 = 2\).

Thus, the second term simplifies to:
\[ 2a \]

### Step 3: Combine the Results
Now we combine the simplified terms:
\[ 4b^2 - 2a \]

### Step 4: Factor Out Common Terms
We can factor out the common factor of \(2\):
\[ 2(2b^2 - a) \]

### Final Answer
The simplified expression is:
\[ 2(2b^2 - a) \]

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