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Chapter 10: Problem 23

Determine the greatest common factor. \(w^{3}, w^{2},\) and \(w\)

Short Answer

Expert verified
The GCF is \(w\).

Step by step solution

01

- List the terms

Identify all given terms: 1. \(w^{3}\), 2. \(w^{2}\), and 3. \(w\).
02

- Determine the factors

For each term, list out the factors. \(w^{3} = w \times w \times w\), \(w^{2} = w \times w\), and \(w = w\).
03

- Identify the common factors

Find the factors common to all terms. Each term has at least one \(w\) as a factor.
04

- Determine the greatest common factor

The greatest common factor (GCF) is the highest degree of the common factor present in each term. In this case, it鈥檚 \(w\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prealgebra
Prealgebra lays the foundation for all future math courses by introducing basic mathematical concepts. It covers arithmetic operations, basic geometry, and introduces algebraic thinking. One significant aspect of prealgebra is working with factors and finding the greatest common factor (GCF). This skill helps you simplify problems and understand relationships between numbers. When solving problems like determining the GCF, prealgebra teaches you to break down expressions into simpler parts, making complex problems easier to handle.
Factors
Factors are the numbers or expressions that divide a given number or expression without leaving a remainder. For example, the factors of 6 are 1, 2, 3, and 6. When working with algebraic expressions like in the exercise, you need to identify the factors of each term. For instance, consider the term \(w^3\). The factors are \(w \times w \times w\). These factors are the building blocks you need to understand to find common factors and the greatest common factor.
Common Factors
Common factors are the factors that two or more numbers or expressions share. To find them, list out all the factors for each term and see which ones appear in all lists. In our example, with the terms \(w^3\), \(w^2\), and \(w\), the factors for each term are:
  • \(w^3 = w \times w \times w\)
  • \(w^2 = w \times w\)
  • \(w = w\)
The common factor here is simply \(w\), which appears in each list of factors.
GCF
The Greatest Common Factor (GCF) is the largest factor that two or more terms share. In the exercise, after listing out the factors and identifying the common factors, we saw that \(w\) is the common factor for \(w^3\), \(w^2\), and \(w\). To find the GCF, take the highest degree of the common factors. Since the common factor is \(w\) and it appears as \(w\) in all terms, the GCF is simply \(w\). Finding the GCF is an essential skill in algebra as it helps simplify expressions and solve equations efficiently.

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