What will be the digit at unit's place of `9^(n)`?

Answer

To find the digit at the unit's place of \( 9^n \), we can analyze the unit's place of \( 9^n \) for different values of \( n \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to determine the unit's place digit of \( 9^n \) for any integer \( n \). The unit's place digit is the last digit of the number.

2. **Identify the Cases**:
The exponent \( n \) can either be odd or even. We will analyze both cases separately.

3. **Case 1: When \( n \) is Odd**:
- Let's take \( n = 1 \):
\[ 9^1 = 9 \]
The unit's place digit is **9**.
- Let's take \( n = 3 \):
\[ 9^3 = 729 \]
The unit's place digit is also **9**.
- Continuing this for other odd values of \( n \) (like \( n = 5, 7, \ldots \)), we find that the unit's place digit remains **9**.

4. **Case 2: When \( n \) is Even**:
- Let's take \( n = 2 \):
\[ 9^2 = 81 \]
The unit's place digit is **1**.
- Let's take \( n = 4 \):
\[ 9^4 = 6561 \]
The unit's place digit is also **1**.
- Continuing this for other even values of \( n \) (like \( n = 6, 8, \ldots \)), we find that the unit's place digit remains **1**.

5. **Conclusion**:
- If \( n \) is odd, the unit's place digit of \( 9^n \) is **9**.
- If \( n \) is even, the unit's place digit of \( 9^n \) is **1**.

### Final Answer:
- The unit's place digit of \( 9^n \) is:
- **9** if \( n \) is odd.
- **1** if \( n \) is even.