Problem 9 - Prealgebra

Question

List all square roots of the given number. If the number has no square roots, write “none”. 144

Step-by-Step Solution

Answer

The square roots of 144 are 12 and -12.

Step 1: Identify the Number

The number provided in the exercise is 144. The problem asks us to find the square roots of this number.

Step 2: Define Square Root Concept

The square root of a number is a value that, when multiplied by itself, gives the original number. If \( x \) is the square root of 144, then \( x^2 = 144 \).

Step 3: Calculate the Positive Square Root

To find the positive square root of 144, we look for a number that multiplied by itself equals 144. Using calculation, \( 12 \times 12 = 144 \), so one square root is 12.

Step 4: Calculate the Negative Square Root

Since squaring a negative number also gives a positive result, we also consider \(-12\), because \((-12) \times (-12) = 144\). Therefore, \(-12\) is also a square root of 144.

Step 5: List all Square Roots

The number 144 has two square roots: 12 and -12.

Key Concepts

Positive and Negative Roots

When we talk about square roots, many students initially think of only the positive value. However, it's important to recognize that every positive number actually has two square roots: one positive and one negative.

• Positive Square Root: This is the non-negative value that results from the square root operation. For the number 144, the positive square root is 12 because 12 times 12 equals 144.
• Negative Square Root: This is the negative value that also satisfies the square root equation. In the case of 144, (-12) also works because (-12) times (-12) is still 144.

This dual characteristic is crucial because it helps us solve equations more comprehensively. Both roots are solutions that should be considered in mathematical problems involving square roots.

Prealgebra Concepts

Before diving into more complex algebra, understanding prealgebra concepts is vital. One of these foundational concepts is square roots. A square root of a number is essentially one of the two equal factors of that number. In prealgebra, we learn basic rules of how numbers interact, especially regarding multiplication and division. Seeing how square roots relate to these operations enhances our ability to solve a broad array of mathematical problems.

• Understanding Squares: A number squared is simply that number multiplied by itself, like 12 squared is 144.
• Recognizing Square Roots: Since squaring is essentially repeated multiplication, finding a square root can be seen as the inverse operation.

Mastering these basic concepts sets the groundwork for algebra, where such operations become core components of more complicated equations.

Basic Arithmetic Operations

Arithmetic operations form the backbone of mathematics. These include addition, subtraction, multiplication, and division. Understanding these allows students to tackle more advanced topics, such as finding square roots. To find the square root of a number, we use multiplication, as this operation helps us check whether a guessed root truly multiplies back to the original number.

• Multiplication Check: For instance, to verify the positive root of 144 is 12, we check if 12 times 12 equals 144. In the same way, for the negative root (-12), we ensure that (-12) times (-12) also equals 144.
• Inverse Operations: Basic arithmetic also teaches us that certain operations have inverses, like multiplication which is counterbalanced by division.

Mastering these simple arithmetic operations is crucial for efficiently handling square root problems and other math challenges.