Multiply `(1)/(2)a^(2)b-(2)/(3)ab^(2)+b` by 6abc:

Options

A. `3a^(2)b^(2)c-4a^(2)b^(3)c-30abc`
B. `3a^(3)b^(3)c-4a^(3)b^(2)c+6abc`
C. `6a^(3)b^(3)+18ab^(2)c+30abc`
D. none of these

Answer

To solve the problem of multiplying \(\frac{1}{2}a^{2}b - \frac{2}{3}ab^{2} + b\) by \(6abc\), we will follow these steps:

### Step 1: Distribute \(6abc\) to each term in the expression
We will multiply each term in the expression \(\frac{1}{2}a^{2}b - \frac{2}{3}ab^{2} + b\) by \(6abc\).

\[ 6abc \left(\frac{1}{2}a^{2}b\right) - 6abc \left(\frac{2}{3}ab^{2}\right) + 6abc(b) \]

### Step 2: Calculate each term separately

1. **First term:**
\[ 6abc \cdot \frac{1}{2}a^{2}b = \frac{6}{2} \cdot a^{2} \cdot b \cdot abc = 3a^{3}b^{2}c \]

2. **Second term:**
\[ 6abc \cdot \left(-\frac{2}{3}ab^{2}\right) = -\frac{6 \cdot 2}{3} \cdot a \cdot b^{2} \cdot abc = -4a^{2}b^{3}c \]

3. **Third term:**
\[ 6abc \cdot b = 6 \cdot a \cdot b \cdot b \cdot c = 6ab^{2}c \]

### Step 3: Combine all the terms
Now we will combine all the terms we calculated:

\[ 3a^{3}b^{2}c - 4a^{2}b^{3}c + 6ab^{2}c \]

### Step 4: Factor out the common term
All terms have a common factor of \(c\):

\[ c(3a^{3}b^{2} - 4a^{2}b^{3} + 6ab^{2}) \]

### Final Answer
The final expression after multiplication and combining like terms is:

\[ c(3a^{3}b^{2} - 4a^{2}b^{3} + 6ab^{2}) \]