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

2. **Substitute \( b \) into the expression**:
Now, substitute \( b \) into the expression \( (a-3)^7 + (b-2)^7 \):
\[ (a-3)^7 + ((5-a)-2)^7 \]
Simplifying \( (5-a)-2 \):
\[ (5-a)-2 = 3-a \]
So, the expression becomes:
\[ (a-3)^7 + (3-a)^7 \]

3. **Recognize the symmetry in the expression**:
Notice that \( (3-a) \) is simply \( -(a-3) \). Thus, we can rewrite the expression:
\[ (a-3)^7 + (-(a-3))^7 \]
Since the exponent is odd (7), we have:
\[ (-(a-3))^7 = -(a-3)^7 \]

4. **Combine the terms**:
Now we can combine the two terms:
\[ (a-3)^7 - (a-3)^7 = 0 \]

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