`16/9 xx (-1(1)/2)^(3)=`

Options

A. -12
B. -6
C. `-8/3`
D. `8/9`

Answer

To solve the expression \( \frac{16}{9} \times \left(-1 \frac{1}{2}\right)^3 \), we will follow these steps:

### Step 1: Convert the mixed number to an improper fraction
The mixed number \(-1 \frac{1}{2}\) can be converted to an improper fraction:
\[ -1 \frac{1}{2} = -\left(1 \times 2 + 1\right) / 2 = -\frac{3}{2} \]

### Step 2: Cube the improper fraction
Now we need to cube \(-\frac{3}{2}\):
\[ \left(-\frac{3}{2}\right)^3 = -\frac{3^3}{2^3} = -\frac{27}{8} \]

### Step 3: Multiply by \(\frac{16}{9}\)
Next, we multiply \(\frac{16}{9}\) by \(-\frac{27}{8}\):
\[ \frac{16}{9} \times -\frac{27}{8} = -\frac{16 \times 27}{9 \times 8} \]

### Step 4: Simplify the multiplication
Now we simplify the multiplication:
\[ - \frac{16 \times 27}{9 \times 8} = -\frac{432}{72} \]

### Step 5: Simplify the fraction
Now we simplify \(-\frac{432}{72}\):
\[ - \frac{432 \div 72}{72 \div 72} = -6 \]

### Final Answer
Thus, the final answer is:
\[ -6 \]