Problem 74 - Prealgebra

Question

Factor each of the following numbers into the product of two numbers, one of which is a perfect square. (Remember from Chapter 1, a perfect square is \(1,4,9,16,25,36, \ldots\) etc. $$20$$

Step-by-Step Solution

Answer

20 = 4 × 5, with 4 being a perfect square.

Step 1: Identify Perfect Squares Less Than 20

List the perfect squares less than 20. They are 1, 4, 9, and 16.

Step 2: Check Each Perfect Square for Factorization

Check if each perfect square can be multiplied by another number to result in 20. Start with the largest perfect square, 16.

Step 3: Test 16 as a Factor

16 is not a factor of 20 as 20/16 is not a whole number. Move to the next perfect square, 9.

Step 4: Test 9 as a Factor

9 is not a factor of 20 as 20/9 is not a whole number. Move to the next perfect square, 4.

Step 5: Test 4 as a Factor

4 is a factor of 20 since 20/4 equals 5, which is a whole number. Thus, the two numbers are 4 (a perfect square) and 5.

Step 6: Conclusion

The number 20 can be factored into the product of the perfect square 4 and the whole number 5.

Key Concepts

Factorization

Factorization is a method used in mathematics to express a number as the product of other numbers, which are referred to as factors. This is particularly useful when solving equations or simplifying expressions. In the context of our example, factorization involves breaking down a number into two parts where one is a perfect square.

• A perfect square is a number that can be expressed as the square of an integer, such as 1, 4, 9, 16, and so on.
• The process of factorization helps simplify complex problems and is a crucial step in algebraic operations.
• When factorizing a number, it's important to start with any known perfect squares to see if they can be factors.

Understanding how to factorize numbers is a foundational skill in prealgebra and higher-level mathematics, as it provides the ability to simplify expressions and solve equations more efficiently.

Prealgebra

Prealgebra is the foundation of all advanced mathematics and focuses on basic arithmetic operations, number theory, and introductory algebraic concepts. It's like the stepping stone into the world of algebra.

• Prealgebra covers topics such as factors, multiples, and integers.
• It introduces students to the idea of simplifying mathematical expressions, such as finding common factors or factorizing numbers.
• This stage is where learners get comfortable understanding the concept of perfect squares and how to use them in operations like factorization.

By mastering prealgebra concepts, students gain confidence to tackle more complex problems in subjects like algebra, geometry, and beyond. This sets a strong mathematical foundation for future learning.

Mathematics

Mathematics encompasses a wide variety of topics including arithmetic, algebra, geometry, and calculus. Among these, factorization is a key concept within arithmetic and algebra that helps simplify calculations.
In our exercise example, factorization is used to decompose 20 into its factors where one factor is a perfect square, i.e., 4 and 5.

• This showcases how math concepts are interconnected and how they build upon one another to solve problems.
• Learning these fundamental math skills, like factorization and the identification of perfect squares, forms a crucial part of mathematical literacy.
• These skills are not only essential for academic success but are also valuable in real-world problem-solving scenarios.

Mathematics provides the tools to engage with analytical thinking and logically solve everyday problems. It's an essential subject that enhances critical thinking and problem-solving skills across various domains.