Multiply :
`2a^(3) - 3a^(2)b` and `-(1)/(2)ab^(2)`

Answer

To multiply the expressions \(2a^3 - 3a^2b\) and \(-\frac{1}{2}ab^2\), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the multiplication:
\[ (2a^3 - 3a^2b) \cdot \left(-\frac{1}{2}ab^2\right) \]

### Step 2: Distribute the terms
Next, we distribute \(-\frac{1}{2}ab^2\) to each term in the first expression:
\[ = 2a^3 \cdot \left(-\frac{1}{2}ab^2\right) + (-3a^2b) \cdot \left(-\frac{1}{2}ab^2\right) \]

### Step 3: Multiply the first term
Now, we calculate the first term:
\[ 2a^3 \cdot \left(-\frac{1}{2}ab^2\right) = -\frac{2}{2} a^{3+1} b^2 = -a^4b^2 \]

### Step 4: Multiply the second term
Next, we calculate the second term:
\[ (-3a^2b) \cdot \left(-\frac{1}{2}ab^2\right) = \frac{3}{2} a^{2+1} b^{1+2} = \frac{3}{2} a^3b^3 \]

### Step 5: Combine the results
Now, we combine the results from both multiplications:
\[ -a^4b^2 + \frac{3}{2}a^3b^3 \]

### Step 6: Factor out the negative sign
Finally, we can factor out the negative sign:
\[ = -a^4b^2 + \frac{3}{2}a^3b^3 \]

### Final Expression
Thus, the final expression is:
\[ \frac{3}{2}a^3b^3 - a^4b^2 \]