A certain pefect square has exactly 4 digits (that is, it is an integer between 1,000 and 9,999). The positive square root of the perfect square must have how many digits?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To determine how many digits the positive square root of a perfect square with exactly four digits has, we can follow these steps:

### Step 1: Identify the range of the perfect square
A perfect square with exactly four digits must be between 1000 and 9999.

### Step 2: Find the square roots of the boundaries
To find the square roots of the boundaries:
- The square root of 1000 is approximately \( \sqrt{1000} \approx 31.62 \).
- The square root of 9999 is approximately \( \sqrt{9999} \approx 99.99 \).

### Step 3: Determine the integer range of the square roots
Since we are looking for perfect squares, we need to consider the integer values:
- The smallest integer greater than \( 31.62 \) is 32.
- The largest integer less than \( 99.99 \) is 99.

### Step 4: Count the digits in the square roots
Now we need to determine how many digits the integers from 32 to 99 have:
- The number 32 has 2 digits.
- The number 99 also has 2 digits.

### Conclusion
Since both the smallest and largest square roots (32 and 99) have 2 digits, we conclude that the positive square root of a perfect square with exactly four digits must have **2 digits**.

### Final Answer
The positive square root of the perfect square must have **2 digits**.

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