Ifx and y are any natural numbers and `x square y=x xx y, " if " x + y " is an even. " = y^2, " if " x + y " is an odd. " x `Deleta` y=x^2, " if " x^y " is an even " = x^2 - y^2, i x^y " is an odd " ( 9 square 11) Delta 4` equals to -

Options

A. 297
B. 9785
C. 9801
D. None of these

Answer

To solve the problem, we need to follow the steps outlined in the video transcript and apply the given rules for the operations defined as "square" and "Delta".

### Step-by-Step Solution:

1. **Identify x and y**:
We have \( x = 9 \) and \( y = 11 \).

2. **Check if \( x + y \) is even or odd**:
Calculate \( x + y = 9 + 11 = 20 \). Since 20 is even, we will use the first condition.

3. **Apply the first condition**:
According to the problem, if \( x + y \) is even, then:
\[ x \square y = x \times y \]
Therefore, we calculate:
\[ 9 \square 11 = 9 \times 11 = 99 \]

4. **Calculate \( x^y \)**:
Now, we need to compute \( 99^4 \) to check if it is even or odd.
Since \( 99 \) is odd, any power of an odd number remains odd. Thus, \( 99^4 \) is odd.

5. **Apply the second condition for Delta**:
Since \( x^y \) is odd, we use the second condition:
\[ x \Delta y = x^2 - y^2 \]
Now calculate \( x^2 \) and \( y^2 \):
\[ x^2 = 99^2 = 9801 \]
\[ y^2 = 4^2 = 16 \]
Therefore:
\[ x \Delta y = 99^2 - 4^2 = 9801 - 16 = 9785 \]

6. **Final answer**:
The final result of \( 9 \Delta 11 \) is:
\[ 9 \Delta 11 = 9785 \]

### Summary:
The answer to the question \( (9 \square 11) \Delta 4 \) is \( 9785 \).

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