The four digit number 8ab9 is a perfect square. The value of `a^(2) + b^2` is

Answer

To solve the problem, we need to find the digits \( a \) and \( b \) in the four-digit number \( 8ab9 \) such that this number is a perfect square. We will then calculate \( a^2 + b^2 \).

### Step-by-step Solution:

1. **Identify the Range of Perfect Squares:**
The number \( 8ab9 \) is a four-digit number that lies between 8000 and 9000. Therefore, we need to find the integer \( n \) such that:
\[ 8000 < n^2 < 9000 \]

2. **Calculate the Square Roots:**
Taking the square root of the boundaries:
\[ \sqrt{8000} \approx 89.44 \quad \text{and} \quad \sqrt{9000} \approx 94.87 \]
This means \( n \) must be an integer between 90 and 94.

3. **Determine Possible Values of \( n \):**
The possible integer values for \( n \) are 90, 91, 92, 93, and 94.

4. **Check Which Values Yield a Perfect Square Ending in 9:**
A perfect square can only end in 9 if the last digit of \( n \) is either 3 or 7. Therefore, we check:
- \( n = 93 \) (since 90, 91, 92, and 94 do not end in 3 or 7)

5. **Calculate \( n^2 \):**
Now we calculate:
\[ n^2 = 93^2 = 8649 \]

6. **Identify \( a \) and \( b \):**
From the number \( 8649 \), we can see that:
- \( a = 6 \)
- \( b = 4 \)

7. **Calculate \( a^2 + b^2 \):**
Now we compute:
\[ a^2 + b^2 = 6^2 + 4^2 = 36 + 16 = 52 \]

### Final Answer:
The value of \( a^2 + b^2 \) is \( 52 \).