Use The Properties Of Real Numbers To Simplify The Following
Use The Properties Of Real Numbers To Simplify The Following
In algebra, the properties of real numbers serve as fundamental tools that facilitate the process of simplifying complex expressions. By understanding and applying these properties, such as the distributive property, the commutative property, and the associative property, students can manipulate algebraic expressions systematically and accurately. This paper demonstrates the process of simplifying three algebraic expressions using the properties of real numbers, illustrating their importance in algebraic problem-solving. Additionally, it discusses the usefulness of these properties in simplifying algebraic expressions and emphasizes the significance of knowing these properties for effective mathematical communication and reasoning.
Mathematical Solution and Explanation
Expression 1: 2a(a – 5) + 4(a – 5)
| Math Work | Property Used |
|---|---|
| 2a(a – 5) + 4(a – 5) | Original expression |
| 2a a – 2a 5 + 4(a – 5) | Distributive property |
| 2a2 – 10a + 4a – 20 | Distributive property (applied to 4(a – 5)) |
| 2a2 – (10a – 4a) – 20 | Combining like terms, showing subtraction as addition of negative |
| 2a2 – 6a – 20 | Like terms |
Expression 2: 2w – 3 + 3(w – 4) – 5(w – 6)
| Math Work | Property Used |
|---|---|
| 2w – 3 + 3w – 12 – 5w + 30 | Distributive property (applied to 3(w – 4) and –5(w – 6)) |
| (2w + 3w – 5w) + (–3 – 12 + 30) | Removing parentheses and grouping like terms |
| (0w) + 15 | Like terms (w terms) summarized, coefficient of w is zero |
| 15 | Constants combined |
Expression 3: 0.05(0.3m + 35n) – 0.8(–0.09n – 22m)
| Math Work | Property Used |
|---|---|
| 0.05 0.3m + 0.05 35n – 0.8 (–0.09n) – 0.8 (–22m) | Distributive property (removing parentheses) |
| 0.015m + 1.75n + 0.072n + 17.6m | Multiplying coefficients accordingly |
| (0.015m + 17.6m) + (1.75n + 0.072n) | Grouping like terms |
| 17.615m + 1.822n | Like terms simplified |
Importance of Properties of Real Numbers in Algebra
The properties of real numbers are crucial in algebra because they provide a structured foundation for manipulating and simplifying expressions. The distributive property, for example, allows one to remove parentheses by multiplying each term within parentheses by a coefficient, thus making expressions more manageable. The like terms property enables combining similar variable terms, which simplifies the expression to its most concise form. The concept of a coefficient—the numerical factor multiplying a variable—is fundamental in understanding how amounts scale within an expression. Without these properties, solving algebra problems would be significantly less efficient and more prone to errors.
Moreover, these properties facilitate the process of simplifying complex expressions, making it easier to solve equations and understand relationships between variables. They support mathematical reasoning by allowing consistent, logical transformations of expressions, which is essential in higher mathematics, science, engineering, and everyday problem-solving. For example, understanding how to remove parentheses using the distributive property enables students to accurately expand and factor expressions, essential skills in algebraic manipulation.
In summary, mastery of the properties of real numbers is vital for developing computational fluency and conceptual understanding in algebra. These properties serve as the building blocks for more advanced mathematical concepts and problem-solving strategies.
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