`a o+ b=(a+b)^(2)-ab`
`a**b=(ab)^(2)-ab`
`a"@"b=(a-b)^(2)-ab`
Find the value of `(((1o+2)**3)"@" 20)`:

Options

A. 151600
B. 141500
C. 161500
D. none of these

Answer

To solve the expression \( (((1o+2)**3)"@" 20) \), we will break it down step by step using the definitions provided for the operations \( o+ \), \( * \), and \( @ \).

### Step 1: Calculate \( 1o+2 \)

Using the definition of \( o+ \):
\[ a o+ b = (a + b)^2 - ab \]
Substituting \( a = 1 \) and \( b = 2 \):
\[ 1 o+ 2 = (1 + 2)^2 - (1 \cdot 2) \]
Calculating:
\[ = 3^2 - 2 = 9 - 2 = 7 \]

### Step 2: Calculate \( (1o+2)^3 \)

Now we need to compute \( 7^3 \):
\[ 7^3 = 7 \cdot 7 \cdot 7 = 343 \]

### Step 3: Calculate \( 343"@"20 \)

Using the definition of \( @ \):
\[ a "@" b = (a - b)^2 - ab \]
Substituting \( a = 343 \) and \( b = 20 \):
\[ 343 "@" 20 = (343 - 20)^2 - (343 \cdot 20) \]
Calculating:
\[ = (323)^2 - (343 \cdot 20) \]
Calculating \( (323)^2 \):
\[ 323^2 = 104329 \]
Calculating \( 343 \cdot 20 \):
\[ 343 \cdot 20 = 6860 \]
Now substituting these values back:
\[ = 104329 - 6860 = 97669 \]

### Final Answer

Thus, the value of \( (((1o+2)**3)"@" 20) \) is:
\[ \boxed{97669} \]