If `a^(**) b=a^(2)+b^(2)-ab` for all natural numbers a and b, then the value of `9^(**) 10` is

Options

A. 181
B. 90
C. 91
D. 182

Answer

To solve the problem where \( a^{**} b = a^2 + b^2 - ab \) for natural numbers \( a \) and \( b \), we need to find the value of \( 9^{**} 10 \).

### Step-by-Step Solution:

1. **Identify the values of \( a \) and \( b \)**:
Here, we have \( a = 9 \) and \( b = 10 \).

2. **Substitute \( a \) and \( b \) into the formula**:
We use the given operation:
\[ 9^{**} 10 = 9^2 + 10^2 - 9 \cdot 10 \]

3. **Calculate \( 9^2 \)**:
\[ 9^2 = 81 \]

4. **Calculate \( 10^2 \)**:
\[ 10^2 = 100 \]

5. **Calculate \( 9 \cdot 10 \)**:
\[ 9 \cdot 10 = 90 \]

6. **Combine the results**:
Now substitute these values back into the equation:
\[ 9^{**} 10 = 81 + 100 - 90 \]

7. **Perform the addition and subtraction**:
First, add \( 81 + 100 \):
\[ 81 + 100 = 181 \]
Then subtract \( 90 \):
\[ 181 - 90 = 91 \]

8. **Final result**:
Thus, the value of \( 9^{**} 10 \) is:
\[ \boxed{91} \]