Correct Answer:
A. 99
The correct answer is 99. The square root of 9801 is the number which, when multiplied by itself, gives 9801. By calculation or recall, we know that 99 × 99 = 9801, so √9801 = 99.
Step‑by‑Step Verification
- Estimate the range: 100 × 100 = 10,000, which is slightly larger than 9,801. This tells us the square root must be just below 100.
- Test 99: 99 × 99 = (100 − 1)² = 100² − 2(100)(1) + 1² = 10,000 − 200 + 1 = 9,801. This matches perfectly.
- Alternate method: You can also factor 9801. It is divisible by 9 (sum of digits 9+8+0+1=18), and 9801 ÷ 9 = 1089. Then 1089 = 33 × 33. So 9801 = 9 × 1089 = 3² × 33² = (3 × 33)² = 99². Hence the root is 99.
Why the Other Options Are Incorrect
- Option 2 (101): 101² = 10,201, which is greater than 9,801.
- Option 3 (103): 103² = 10,609, far larger than 9,801.
- Option 4 (109): 109² = 11,881, also much larger than the given number.
All other options produce a square greater than 9,801, while 99² exactly equals the given value. Therefore, the only correct choice is 99 (Option 1).