Use The Properties Of Real Numbers To Simplify.

Use the properties of real numbers to simplify the following expressions

Read the following instructions in order to complete this assignment, and review the example of how to complete the math required for this assignment: Use the properties of real numbers to simplify the following expressions: 2a(a – 5) + 4(a – w – 3 + 3(w – 4) – 5(w – .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, please 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 (Do not write definitions for the words; use them appropriately in sentences describing your math work.): Simplify, Like terms, Coefficient, Distribution, Removing parentheses. For information regarding APA samples and tutorials, visit the Ashford Writing Center, within the Learning Resources tab on the left navigation toolbar. Carefully review the Grading Rubric for the criteria that will be used to evaluate your assignment.

Paper For Above instruction

The goal of this paper is to demonstrate the process of simplifying complex algebraic expressions by applying the properties of real numbers, while also discussing their importance and utility in algebra. The specific expression to simplify is: 2a(a – 5) + 4(a – w – 3 + 3(w – 4) – 5(w – .05(0.3m + 35n) – 0.8(-0.09n – 22m)). The approach involves breaking down each component, distributing coefficients, removing parentheses, and combining like terms systematically. Additionally, incorporating relevant math vocabulary in context enriches the explanation. This example illustrates how understanding such properties simplifies complex expressions, making algebra manageable and logical.

Initially, observe the first term, 2a(a – 5). To distribute the 2a, multiply it across the binomial, yielding 2a a = 2a² and 2a (–5) = –10a. The property used here is the distributive property, which allows us to multiply a coefficient across terms within parentheses, removing parentheses in the process. Next, consider the second part, 4(a – w – 3 + 3(w – 4) – 5(w – .05(0.3m + 35n) – 0.8(–0.09n – 22m))). Inside the brackets, we first focus on 3(w – 4). Applying distribution, 3 w = 3w and 3 (–4) = –12. Then, we move to –5(w – .05(0.3m + 35n) – 0.8(–0.09n – 22m)).

Within this, the terms .05(0.3m + 35n) and 0.8(–0.09n – 22m) need to be distributed as well, moving step-by-step. For .05(0.3m + 35n), the multiplication gives 0.05 0.3m = 0.015m, and 0.05 35n = 1.75n. For 0.8(–0.09n – 22m), multiplying each term by 0.8 gives –0.072n and –17.6m, respectively. After these distributions, the expression inside the brackets contains multiple similar terms involving m and n, which we will combine.

Now, the expression with all distributed constants becomes more manageable, and similar terms are identified as like terms. For instance, grouping all terms involving m: 0.015m and –17.6m, and those involving n: 1.75n and –0.072n, allows us to combine these like terms by adding or subtracting their coefficients. This process is guided by the associative and commutative properties of addition and multiplication, which facilitate simplification.

The next step involves multiplying the outer coefficient 4 by each term inside the brackets, continuing the distribution process for all remaining terms. Be sure to keep track of the signs and coefficients, and to simplify at each step, ensuring that similar algebraic terms are like terms. This systematic approach illustrates the importance of understanding the properties of real numbers, as they provide a logical framework enabling us to simplify complex expressions correctly.

Throughout this process, the distributive property is essential for removing parentheses and spreading coefficients across multiple terms. Recognizing like terms allows for effective simplification, which ultimately results in a more concise and manageable algebraic expression. This exercise emphasizes that a solid grasp of these properties enhances efficiency, accuracy, and clarity in algebraic manipulations.

The properties of real numbers serve as foundational tools in algebra, allowing mathematicians and students alike to simplify and understand complex expressions systematically. Their usefulness extends beyond basic calculations; they underpin the structure of algebra and support problem-solving strategies. In particular, the distributive property and recognition of like terms enable us to convert intricate expressions into simpler forms, facilitating easier interpretation and further mathematical operations.

References

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