if the operation `o.` is defined for all real numbers a and b by the relation ` o. =(a^2 b )/(3)`
then 2 `o. {3o.(-1)}`

Options

A. 2
B. 4
C. `-4`
D. `-2`

Answer

To solve the problem step by step, we will evaluate the operation defined by \( a \, o. \, b = \frac{a^2 b}{3} \). We need to find the value of \( 2 \, o. \, (3 \, o. \, (-1)) \).

### Step 1: Evaluate \( 3 \, o. \, (-1) \)

Using the operation definition:
\[ 3 \, o. \, (-1) = \frac{3^2 \cdot (-1)}{3} \]

Calculating \( 3^2 \):
\[ 3^2 = 9 \]

Now substituting back into the equation:
\[ 3 \, o. \, (-1) = \frac{9 \cdot (-1)}{3} = \frac{-9}{3} = -3 \]

### Step 2: Evaluate \( 2 \, o. \, (-3) \)

Now we need to use the result from Step 1 to evaluate \( 2 \, o. \, (-3) \):
\[ 2 \, o. \, (-3) = \frac{2^2 \cdot (-3)}{3} \]

Calculating \( 2^2 \):
\[ 2^2 = 4 \]

Now substituting back into the equation:
\[ 2 \, o. \, (-3) = \frac{4 \cdot (-3)}{3} = \frac{-12}{3} = -4 \]

### Final Result

Thus, the value of \( 2 \, o. \, (3 \, o. \, (-1)) \) is:
\[ \boxed{-4} \]