If `f(A, B)=A#B=A^(4)-B^(3)+A^(2)-B+AB` then find the value of `f(3, 4)`

Answer

To find the value of \( f(3, 4) \) using the function defined as \( f(A, B) = A^4 - B^3 + A^2 - B + AB \), we will substitute \( A = 3 \) and \( B = 4 \) into the function and simplify step by step.

### Step-by-Step Solution:

1. **Substitute the values of A and B**:
\[ f(3, 4) = 3^4 - 4^3 + 3^2 - 4 + 3 \times 4 \]

2. **Calculate \( 3^4 \)**:
\[ 3^4 = 81 \]

3. **Calculate \( 4^3 \)**:
\[ 4^3 = 64 \]

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

5. **Calculate \( 3 \times 4 \)**:
\[ 3 \times 4 = 12 \]

6. **Substitute these values back into the function**:
\[ f(3, 4) = 81 - 64 + 9 - 4 + 12 \]

7. **Perform the arithmetic step-by-step**:
- First, calculate \( 81 - 64 \):
\[ 81 - 64 = 17 \]
- Next, add \( 9 \):
\[ 17 + 9 = 26 \]
- Then, subtract \( 4 \):
\[ 26 - 4 = 22 \]
- Finally, add \( 12 \):
\[ 22 + 12 = 34 \]

8. **Final result**:
\[ f(3, 4) = 34 \]

### Final Answer:
The value of \( f(3, 4) \) is \( 34 \).