Question

A certain perfect square has exactly 4 digits (that is, it \(\text { is an integer between } 1,000 \text { and } 9,999) .\) The positive square root of the perfect square must have how many digits? F. 1 G. 2 H. 3 I. 4 K. Cannot be determined from the given information

Short answer

Answer: G. 2

Step-by-step solution

Determine the range of 4-digit perfect squares

The problem states that the perfect square is a 4-digit integer. A 4-digit integer is any integer between 1000 and 9999. Let's represent the lower bound (smallest 4-digit perfect square) as \(L^2\) and the upper bound (largest 4-digit perfect square) as \(U^2\). So, we have: \(1000 \le L^2 \le U^2 \le 9999\)

Find the smallest and largest possible values for L and U

Now, we will find the smallest and the largest integer values of L and U by taking the square root of the lower and upper bounds. For L, we have: \(L = \sqrt{1000} \approx 31.62\) Since L must be an integer, we must round up to get the smallest possible value for L. \(L = 32\) For U, we have: \(U = \sqrt{9999} \approx 99.99\) In this case, we round down to get the largest possible value for U. \(U = 99\)

Analyze the number of digits for L and U

Now let's look at the number of digits in the positive square root, L and U. L has 2 digits (32) and U also has 2 digits (99). Therefore, the positive square root of any 4-digit perfect square will have 2 digits.

Choose the correct answer choice

Based on our analysis, the positive square root of a 4-digit perfect square has exactly 2 digits. So the correct answer is: G. 2

Key concepts

Square Root

Understanding what a square root is can be a stepping stone into grasping more complex mathematical concepts. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, if we take the number 9, its square root is 3 because \(3 \times 3 = 9\). It's akin to asking, 'Which number, when squared, gives me the original number?'

When dealing with perfect squares, these are numbers whose square roots are integers. In other words, they are 'neatly' square without any fractions or decimals hanging around. For perfect squares with 4 digits, they fall within an integer range and have a square root that is an integral number with a specific digit count. The square roots themselves aren't always obvious, so we often have to use approximation methods or tools like a calculator to find them, especially when the numbers get larger.

Integer Range

An integer range is essentially a set of consecutive whole numbers starting from one specific integer and ending at another. For our specific exercise, we're dealing with the integer range of perfect squares between 1,000 and 9,999. Here, each of those perfect squares arose from squaring integers within a smaller, definite range. These squares don't just pop up from anywhere; they're each linked to a 'root' integer.

By considering the square roots of the extremes of our 4-digit range (the smallest and largest 4-digit perfect squares), we get to define a narrower band of integers. These integers, when squared, will stay true to our initial criterion: yielding perfect squares that form the 4-digit club. It's also important to observe that for any given range, the squaring function will widen the gap between consecutive numbers. This behavior helps us understand why a 4-digit perfect square must originate from a 2-digit square root.

Digit Count

Digit count is a simple yet essential concept in mathematics, referring to the number of digits that make up a number. It's a fundamental feature of our decimal number system. In our discussion on perfect squares, the digit count directly associates with the size of the square root. The digit count can be viewed as a clue to the number's magnitude.

For example, any 2-digit positive square root will result in at least a 3-digit perfect square (\(10^2 = 100\)) but can reach a maximum of a 4-digit number (\(99^2 = 9801\)). Therefore, knowing the digit count of the square root gives us a quick way to determine the range of possible perfect squares. In terms of providing clarity for students, understanding that a single-digit square root cannot produce a 4-digit perfect square, and a 3-digit square root overshoots into 5 or more digits, is an excellent strategy to simplify the problem.