Simplifying Expressions Using Properties Of Real Numbers

Simplifying Expressions Using Properties of Real Numbers

To complete this assignment, I will simplify the given algebraic expressions using properties of real numbers, document each step methodically, and discuss the importance of these properties in algebra. The expressions are:

• 2a(a – 5) + 4(a – 5)
• 2w – 3 + 3(w – 4) – 5(w – 6)
• 0.05(0.3m + 35n) – 0.8(-0.09n – 22m)

Solution to the Problems and Explanation of Mathematical Properties

1. Simplifying 2a(a – 5) + 4(a – 5)

Step 1: Apply the distributive property to remove parentheses:

Step 2: Simplify each term:

Step 3: Combine like terms (–10a + 4a):

Final expression: 2a^2 – 6a

2. Simplifying 2w – 3 + 3(w – 4) – 5(w – 6)

Step 1: Use distribution to remove parentheses:

Step 2: Now, combine like terms:

Step 3: Simplify both groups:

Final expression: 15

3. Simplifying 0.05(0.3m + 35n) – 0.8(-0.09n – 22m)

Step 1: Use distribution to remove parentheses:

Step 2: Simplify each term:

Step 3: Combine like terms:

Step 4: Simplify sums:

Final expression: 17.615m + 1.822n

Discussion of Properties of Real Numbers and Their Importance in Algebra

The properties of real numbers are foundational in algebra because they allow for systematic manipulation and simplification of expressions. The distributive property, for example, is essential for removing parentheses, thereby enabling us to expand expressions into simpler, like terms, which can then be combined to reduce the overall complexity of the problem. This is particularly useful when dealing with coefficients that multiply variables, as it allows us to distribute the coefficient across each term inside parentheses.

Similarly, the associative and commutative properties help in arranging terms in a more convenient order for combining like terms. The distributive property is instrumental in removing parentheses and is fundamental when dealing with polynomials and algebraic expressions. Recognizing and applying these properties enable students and mathematicians to perform algebraic operations efficiently, enhancing problem-solving skills and logical reasoning, which are crucial for advanced mathematics and real-world applications.

Furthermore, understanding the coefficients—the numerical parts of terms—helps in quantifying the influence of variables within expressions. Manipulating these coefficients through distribution and combination enables the simplification process, which is often necessary for solving equations, optimizing functions, or modeling real-world phenomena. Mastery of these properties and concepts thus facilitates a deeper comprehension of algebra and supports the development of mathematical thinking.

Conclusion

By carefully applying properties of real numbers—particularly the distributive, associative, and commutative properties—we can effectively simplify algebraic expressions. These properties serve as essential tools for removing parentheses, combining like terms, and manipulating coefficients, thereby transforming complex expressions into more manageable forms. Mastery of these properties not only simplifies algebraic work but also enhances logical reasoning and problem-solving abilities, which are vital in both academic and practical contexts.

References

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