Equivalent Expression Of 3(a+5) A Step-by-Step Solution

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Algebraic expressions form the bedrock of mathematics, serving as a symbolic language to represent mathematical relationships. Among the fundamental operations in algebra is the distributive property, a powerful tool for simplifying and manipulating expressions. In this article, we delve into the expression 3(a+5), exploring its equivalent forms and illuminating the underlying principles that govern algebraic transformations. Our focus will be on demonstrating the equivalence of the expression 3(a+5) with one of the provided options, ensuring a thorough understanding of the distributive property and its applications. This exploration is crucial not only for mastering algebraic manipulations but also for building a solid foundation for more advanced mathematical concepts. Understanding how to correctly apply the distributive property is essential for simplifying complex equations, solving for unknowns, and making accurate predictions in various mathematical and real-world scenarios. This skill is not just theoretical; it has practical implications in fields ranging from engineering and physics to economics and computer science.

The expression 3(a+5) presents a scenario where a constant, 3, is multiplied by a binomial, (a+5). The distributive property comes into play here, dictating how we handle this multiplication. This property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. Mathematically, this can be expressed as a(b+c) = ab + ac. Applying this to our expression, we need to multiply 3 by both 'a' and '5'. This process involves distributing the 3 across the terms inside the parentheses. The ability to accurately apply the distributive property is a cornerstone of algebraic manipulation, enabling the simplification of complex expressions and the solution of equations. Mastering this property not only enhances one's mathematical toolkit but also cultivates a deeper understanding of the structure and behavior of algebraic expressions. It is a skill that transcends the classroom, finding relevance in diverse fields where mathematical modeling and problem-solving are paramount.

To simplify 3(a+5), we apply the distributive property by multiplying 3 by each term inside the parentheses. This means we multiply 3 by 'a', resulting in 3a, and then multiply 3 by 5, resulting in 15. The expression then becomes 3a + 15. This step-by-step application of the distributive property is crucial for ensuring accuracy and clarity in algebraic manipulations. Each term within the parentheses must be multiplied by the constant outside, and the resulting products are then combined. This process not only simplifies the expression but also reveals its underlying structure, making it easier to work with in subsequent calculations or problem-solving scenarios. The distributive property is not merely a mechanical rule; it is a fundamental principle that reflects the way multiplication interacts with addition and subtraction. Understanding this interaction is key to developing a strong algebraic intuition and the ability to confidently tackle more complex mathematical problems. By mastering the distributive property, students gain a powerful tool for simplifying expressions, solving equations, and ultimately, advancing their mathematical proficiency.

Now, let's examine the provided answer choices in light of our simplification:

  • A. 3+a+5: This expression simply adds the terms together without applying the distributive property correctly.
  • B. 3a+8: This expression incorrectly adds 3 and 5 before multiplying by 'a'.
  • C. 3a+5: This expression only multiplies 3 by 'a' and not by 5.
  • D. 3a+15: This expression correctly applies the distributive property, multiplying 3 by both 'a' and 5.

By carefully evaluating each option, we can clearly see that only option D, 3a+15, accurately represents the simplified form of the original expression. This process of elimination and verification is a crucial skill in mathematics, ensuring that the chosen answer aligns with both the mathematical principles and the specific problem requirements. Evaluating answer choices not only helps in identifying the correct solution but also reinforces understanding of the underlying concepts and potential pitfalls. It is a methodical approach that promotes accuracy and confidence in problem-solving, fostering a deeper appreciation for the nuances of mathematical reasoning. This skill is particularly valuable in standardized testing scenarios, where careful analysis and evaluation are essential for achieving success.

Based on our application of the distributive property, we've determined that 3(a+5) is equivalent to 3a+15. This equivalence highlights the importance of correctly applying algebraic rules to simplify expressions. The distributive property, as demonstrated, allows us to rewrite expressions in a more manageable form, which is crucial for solving equations and tackling more complex mathematical problems. Understanding and mastering this property is not just about getting the right answer; it's about developing a deeper comprehension of how mathematical operations interact and how expressions can be manipulated to reveal their underlying structure. This understanding forms a solid foundation for further mathematical exploration and problem-solving, empowering students to approach challenges with confidence and precision. The ability to recognize and apply equivalent expressions is a key skill in algebra and beyond, enabling mathematicians, scientists, and engineers to effectively model and solve real-world problems.

In conclusion, the expression 3(a+5) is equivalent to 3a+15, as derived through the correct application of the distributive property. This exercise underscores the significance of understanding and applying fundamental algebraic principles. Mastering the distributive property is essential for simplifying expressions, solving equations, and building a strong foundation in mathematics. It is a skill that extends beyond the classroom, finding application in various fields that rely on mathematical modeling and problem-solving. By understanding how to correctly distribute multiplication over addition, students gain a powerful tool for manipulating algebraic expressions and unlocking their hidden potential. This mastery not only enhances their mathematical abilities but also cultivates a deeper appreciation for the elegance and precision of mathematical reasoning. The journey through algebraic expressions and properties is a rewarding one, leading to a greater understanding of the mathematical world and its applications.