`a"@"b=|a-b|`
`a**b=ab`
`a#b=|a^(2)-b^(2)|`
`aDeltab=a^(2)+b^(2)`
where `a, b in R and a neb` and the real algebraic operations are unchanged.
The value of the expression `[(a#b)div(a"@"b)]^(2)-2(a**b):`

Options

A. `a#b`
B. `aDeltab`
C. `(a#b)(a"@"b)`
D. can't be detetmined

Answer

To solve the expression \([(a \# b) \div (a @ b)]^2 - 2(a ** b)\), we will follow these steps:

### Step 1: Identify the operations
We have the following operations defined:
- \(a @ b = |a - b|\)
- \(a ** b = ab\)
- \(a \# b = |a^2 - b^2|\)
- \(a \Delta b = a^2 + b^2\)

### Step 2: Calculate \(a \# b\)
Using the definition of the operation \(\#\):
\[ a \# b = |a^2 - b^2| \]
This can be factored as:
\[ |a^2 - b^2| = |(a - b)(a + b)| \]

### Step 3: Calculate \(a @ b\)
Using the definition of the operation \(@\):
\[ a @ b = |a - b| \]

### Step 4: Calculate \((a \# b) \div (a @ b)\)
Now we can substitute the results from Steps 2 and 3 into the expression:
\[ \frac{a \# b}{a @ b} = \frac{|(a - b)(a + b)|}{|a - b|} \]
Since \(a \neq b\), we can simplify this:
\[ = |a + b| \]

### Step 5: Square the result
Now we square the result from Step 4:
\[ \left(\frac{a \# b}{a @ b}\right)^2 = (|a + b|)^2 = (a + b)^2 \]

### Step 6: Calculate \(2(a ** b)\)
Using the definition of the operation \(**\):
\[ 2(a ** b) = 2(ab) \]

### Step 7: Combine the results
Now we substitute back into the main expression:
\[ (a + b)^2 - 2(ab) \]

### Step 8: Recognize the identity
Notice that \((a + b)^2 - 2ab\) is the expansion of:
\[ (a - b)^2 \]

### Final Result
Thus, the value of the expression is:
\[ (a - b)^2 \]