`a o+ b=(a+b)^(2)-ab`
`a**b=(ab)^(2)-ab`
`a"@"b=(a-b)^(2)-ab`
Find the value of `(5o+6)-(6"@"3)+(2**4)`

Options

A. 156
B. 256
C. 1056
D. none of these

Answer

To solve the expression \((5o+6)-(6"@3)+(2**4)\), we need to evaluate each binary operation step by step.

### Step 1: Calculate \(5o+6\)

The operation \(a o+ b\) is defined as:
\[ a o+ b = (a + b)^2 - ab \]
Substituting \(a = 5\) and \(b = 6\):
\[ 5 o+ 6 = (5 + 6)^2 - 5 \cdot 6 \]
Calculating:
\[ (5 + 6) = 11 \quad \text{and} \quad 11^2 = 121 \]
Now, compute \(5 \cdot 6\):
\[ 5 \cdot 6 = 30 \]
Thus:
\[ 5 o+ 6 = 121 - 30 = 91 \]

### Step 2: Calculate \(6"@3\)

The operation \(a"@b\) is defined as:
\[ a"@b = (a - b)^2 - ab \]
Substituting \(a = 6\) and \(b = 3\):
\[ 6"@3 = (6 - 3)^2 - 6 \cdot 3 \]
Calculating:
\[ (6 - 3) = 3 \quad \text{and} \quad 3^2 = 9 \]
Now, compute \(6 \cdot 3\):
\[ 6 \cdot 3 = 18 \]
Thus:
\[ 6"@3 = 9 - 18 = -9 \]

### Step 3: Calculate \(2**4\)

The operation \(a**b\) is defined as:
\[ a**b = (ab)^2 - ab \]
Substituting \(a = 2\) and \(b = 4\):
\[ 2**4 = (2 \cdot 4)^2 - 2 \cdot 4 \]
Calculating:
\[ 2 \cdot 4 = 8 \quad \text{and} \quad 8^2 = 64 \]
Now, compute \(2 \cdot 4\):
\[ 2 \cdot 4 = 8 \]
Thus:
\[ 2**4 = 64 - 8 = 56 \]

### Step 4: Combine the results

Now we can substitute back into the original expression:
\[ (5o+6) - (6"@3) + (2**4) = 91 - (-9) + 56 \]
This simplifies to:
\[ 91 + 9 + 56 = 156 \]

### Final Answer
The value of \((5o+6)-(6"@3)+(2**4)\) is \(\boxed{156}\).