Problem 50 - Algebra 1: Concepts and Skills

Question

Determine whether the number is a perfect square. $$ 10,000 $$

Step-by-Step Solution

Answer

Yes, 10,000 is a perfect square. This is because its square root is 100, which is a whole number.

Step 1: Determining the square root

The first step is to calculate the square root of 10,000. In mathematical terms, this would be represented as \( \sqrt{10,000} \).

Step 2: Check if the result is an integer

After calculating the square root, we now need to determine if that result is a whole number. A whole number is an integer that is not a fraction. It can be written without a fractional or decimal component.

Step 3: Is the number a perfect square?

If the square root of 10,000 is a whole number (integer), then we can conclude that 10,000 is a perfect square. If the square root is not an integer, then 10,000 is not a perfect square.

Key Concepts

Square Root

A square root is a value that, when multiplied by itself, gives the original number. For example, when we take the square root of 10,000, we are looking for a number that, when multiplied by itself, equals 10,000.
In mathematical terms, the square root of a number is expressed as \( \sqrt{n} \). The square symbol signifies that we are performing this operation. Calculating a square root requires you to find the number which, squared (multiplied by itself), returns the starting figure we began with.
To determine if 10,000 is a perfect square, we find its square root: \( \sqrt{10,000} = 100 \). This means 100 times 100 equals 10,000.

Integer

An integer is a whole number that does not have a fractional or decimal component. Integers can be positive, negative, or zero, such as -3, 0, and 7.
In the context of identifying a perfect square, the square root of the number must be an integer. If it is an integer, then we can confirm that the original number is, indeed, a perfect square.
From our calculation, the square root of 10,000 is 100. Since 100 is a whole number without any decimals or fractions, it qualifies as an integer. Hence, 10,000 meets the condition for being a perfect square.

Whole Number

Whole numbers are non-negative numbers without fractions or decimals. They start from 0 and include all positive integers like 1, 2, 3, and so on.
It’s important to understand that while all whole numbers are integers, not all integers are whole numbers, as integers can also be negative.
Since 100 has no fractional part and is non-negative, it is a whole number. This confirms that the square root of 10,000 is both an integer and a whole number. Therefore, 10,000 is a perfect square.