If(a+b)=5,then the value of `(a-3)^(7)+(b-2)^(7)` is:

Options

A. `2^(7)`
B. `3^(7)`
C. 1
D. 0

Answer

To solve the problem, we need to find the value of \((a-3)^7 + (b-2)^7\) given that \(a + b = 5\).

### Step-by-Step Solution:

1. **Express \(a\) in terms of \(b\)**:
Since we know \(a + b = 5\), we can express \(a\) as:
\[ a = 5 - b \]

2. **Substitute \(a\) into the expression**:
We need to substitute \(a\) into the expression \((a-3)^7 + (b-2)^7\):
\[ (a-3)^7 = (5-b-3)^7 = (2-b)^7 \]
Therefore, the expression becomes:
\[ (2-b)^7 + (b-2)^7 \]

3. **Recognize the symmetry**:
Notice that \((2-b)\) and \((b-2)\) are negatives of each other:
\[ (b-2) = -(2-b) \]
Hence, we can rewrite the expression:
\[ (2-b)^7 + (b-2)^7 = (2-b)^7 + [-(2-b)]^7 \]

4. **Apply the property of odd powers**:
Since the exponent 7 is odd, we have:
\[ [-(2-b)]^7 = -(2-b)^7 \]
Therefore:
\[ (2-b)^7 + (b-2)^7 = (2-b)^7 - (2-b)^7 = 0 \]

5. **Conclusion**:
The value of \((a-3)^7 + (b-2)^7\) is:
\[ \boxed{0} \]