Use The Properties Of Real Numbers To Simplify 165845

Use The Properties Of Real Numbers To Simplify The Following

Use the properties of real numbers to simplify the following expressions: 2a(a – 5) + 4(a – 5); 2w – 3 + 3(w – 4) – 5(w – 6); 0.05(0.3m + 35n) – 0.8(-0.09n – 22m). Write a two to three page paper that is formatted in APA style and according to the Math Writing Guide. Format your math work as shown in the Instructor Guidance and be concise in your reasoning. In the body of your essay, please make sure to include: your solution to the above problem, making sure to include all mathematical work. Plan the logic necessary to complete the problem before you begin writing. For examples of the math required for this assignment, review Elementary and Intermediate Algebra and the example of how to complete the math required for this assignment.

Show every step of the process of simplifying and identify which property of real numbers was used in each step of your work. Please include your math work on the left; the properties used on the right. A discussion of why the properties of real numbers are important to know when working with algebra. In what ways are they useful for simplifying algebraic expressions? The incorporation of the following five math vocabulary words into the text of your paper. Use bold font to emphasize the words in your writing (Simplify, Like terms, Coefficient, Distribution, Removing parentheses). Do not write definitions for the words; use them appropriately in sentences describing your math work.

Paper For Above instruction

Algebraic expressions are foundational to higher mathematics, and understanding the properties of real numbers facilitates effective manipulation and simplification of these expressions. The properties of real numbers, including the distributive, commutative, associative, identity, and zero properties, are essential tools that provide a structured approach to simplifying complex algebraic expressions. This paper demonstrates the application of these properties through detailed work on the given expressions, illustrating how they enable a step-by-step reduction to simpler forms, thus making solving and understanding algebraic problems more manageable.

Firstly, consider the expression 2a(a – 5) + 4(a – 5). The initial step involves applying the Distribution property to remove parentheses, allowing each term to be multiplied by the coefficient outside. The expression becomes 2a a – 2a 5 + 4 a – 4 5. The Coefficient of each term is identified in these multiplications. Next, we combine like terms, which involves a process of Simplify by grouping terms with the same variable. The terms 2a a (which simplifies to 2a^2) and 4 a are Like terms, as they contain the same variable to the same power. The constant terms, -10 and -20, are also combined if applicable. The resulting expression simplifies to 2a^2 – 10a + 4a – 20. Further Simplify by combining -10a + 4a to get -6a. The complete simplified form is 2a^2 – 6a – 20.

Similarly, for the expression 2w – 3 + 3(w – 4) – 5(w – 6), the first step is to apply the Distribution property to remove parentheses: 2w – 3 + 3 w – 3 4 – 5 w + 5 6. Simplify each term: 2w – 3 + 3w – 12 – 5w + 30. Now, combine the Like terms. The terms involving w are 2w + 3w – 5w, which simplifies to 0w or 0. The constants are -3 – 12 + 30, which sum to 15. The fully Simplifyed expression is 15.

For the expression involving decimals, 0.05(0.3m + 35n) – 0.8(-0.09n – 22m), begin by applying Distribution: 0.05 0.3m + 0.05 35n – 0.8 -0.09n – 0.8 -22m. Calculate each term: 0.015m + 1.75n + 0.072n + 17.6m. Next, combine Like terms: the m terms are 0.015m + 17.6m which sum to 17.615m. The n terms sum to 1.75n + 0.072n, totaling 1.822n. The final Simplifyed expression is 17.615m + 1.822n.

The properties of real numbers are vital because they provide the logical foundation for consistent algebraic manipulation. They ensure that the process of Removing parentheses through Distribution and Like terms consolidation is rigorous and reliable. These properties facilitate Simplifying complex expressions, leading to clearer understanding, solving for variables, and verifying solutions. Mastery of these properties accelerates problem-solving and enhances accuracy in algebra, which are crucial skills both academically and practically in fields involving quantitative reasoning. The systematic application of properties underscores the importance of a structured approach, leading to efficient mathematical communication and problem-solving.

References

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