Problem 50 - Prealgebra

Question

How are square roots related to factors? Give an example of a number between 100 and 200 whose square root is a whole number and an example of a number between 100 and 200 whose square root is a decimal that does not terminate.

Step-by-Step Solution

Answer

144 is a perfect square with an integer root, while 150's root is a non-terminating decimal.

Step 1: Understanding Square Roots and Factors

A square root of a number is a value that, when multiplied by itself, gives the original number. Factors are numbers that divide the original number without leaving a remainder. A perfect square has integer square roots and multiple factors.

Step 2: Choosing an Example of a Perfect Square

We need to find a number between 100 and 200 whose square root is a whole number. A perfect square will help. Let's consider 144. Check if its square root is a whole number by performing the calculation: \( \sqrt{144} = 12 \). Since 12 is an integer, 144 is a perfect square.

Step 3: Choosing an Example of a Non-Perfect Square

We need an example of a number between 100 and 200 whose square root is a decimal that does not terminate (indicating it's not a perfect square). Let's choose 150. Calculate its square root: \( \sqrt{150} \approx 12.247 \). The decimal neither repeats nor terminates.

Key Concepts

Factors

In mathematics, factors are numbers that can be multiplied together to obtain another number. For example, in the equation \(3 \times 4 = 12\), both 3 and 4 are factors of 12. Understanding factors is crucial when working with square roots, especially when determining if a number is a perfect square. A perfect square has factors that pair up perfectly because it can be expressed as the square of an integer.

• Any whole number is divisible by its factors without leaving a remainder.
• For instance, the number 144 has factors like 12 because \(12 \times 12 = 144\).

Recognizing factors helps determine if a number is a perfect square, as it indicates symmetrical pairing necessary for whole number square roots.

Perfect Squares

Perfect squares are numbers that result from multiplying an integer by itself. This means their square roots are always whole numbers. Numbers like 144, 169, and 196 are good examples of perfect squares between 100 and 200. When you calculate the square root of these numbers, you find integers:

• \( \sqrt{144} = 12 \)
• \( \sqrt{169} = 13 \)
• \( \sqrt{196} = 14 \)

Perfect squares are helpful in mathematics, as they provide a clear symmetry that can simplify complex calculations. Recognizing these numbers can streamline the process of rooting and factoring.

Integers

Integers include all whole numbers, both positive and negative, along with zero. Square roots of perfect squares will always result in an integer. For instance, taking the square root of 144 gives us 12, which is an integer. This property of producing whole numbers is essential in many mathematical applications, particularly in geometry and algebra.

• Number 0 is also an integer and plays a central role in integer operations.
• Decimals and fractions are not considered integers.

Understanding integers is vital when working with square roots since determining whether the result is an integer or not can indicate if you are working with a perfect square.

Non-Terminating Decimals

Non-terminating decimals are decimals that go on forever without repeating. These contrast with terminating decimals, which stop after a finite number of digits. When you take the square root of non-perfect squares, you encounter non-terminating decimals. For example, the square root of 150 (which is not a perfect square) is approximately 12.247, and the decimal continues indefinitely.

• These decimals do not revert to a repeating cycle but go on uniquely.
• Non-terminating decimals make irrational numbers when the decimals do not repeat.

Dealing with these types of decimals is common in calculus and higher-level mathematics, where precise approximations are often required. Recognizing these non-terminating decimals is important for understanding the nature of numbers that aren't perfect squares.