Mastering Distributive Property Multiplying 5ab³ By (4a² + 3ab + 2b²)
Introduction to the Distributive Property
In the realm of mathematics, the distributive property stands as a cornerstone principle, enabling us to simplify complex expressions and equations with ease. It's a fundamental concept that bridges the gap between addition and multiplication, providing a systematic approach to handle expressions involving parentheses. At its core, the distributive property dictates that multiplying a single term by an expression enclosed in parentheses is equivalent to multiplying that term by each individual term within the parentheses and then summing the results. This powerful tool is not just a theoretical construct; it's a practical technique widely used in algebra, calculus, and various other branches of mathematics. This article delves into the intricacies of the distributive property, illustrating its application with a detailed example: multiplying the monomial 5ab³ by the trinomial (4a² + 3ab + 2b²). Through a step-by-step breakdown, we aim to provide a comprehensive understanding of this property, empowering you to tackle similar mathematical challenges with confidence and precision. The distributive property essentially states that for any numbers a, b, and c, the equation a(b + c) = ab + ac holds true. This simple yet profound concept is the foundation for expanding expressions and simplifying equations. Understanding and mastering the distributive property is crucial for success in algebra and beyond, as it forms the basis for more advanced mathematical operations and problem-solving techniques. From simplifying algebraic expressions to solving complex equations, the distributive property is an indispensable tool in the mathematician's arsenal.
Problem Statement: Multiplying 5ab³ by (4a² + 3ab + 2b²)
Let's consider the specific problem at hand: multiplying the monomial 5ab³ by the trinomial (4a² + 3ab + 2b²) using the distributive property. This exercise serves as an excellent example to illustrate the practical application of the distributive property in algebraic expressions. The expression 5ab³(4a² + 3ab + 2b²) presents a classic scenario where the distributive property is not just useful, but essential for simplifying the expression. Without it, we would be left with an unexpanded form, hindering further calculations or analysis. Our goal is to systematically apply the distributive property, breaking down the multiplication into manageable steps. This involves multiplying the term outside the parentheses, 5ab³, by each term inside the parentheses individually: 4a², 3ab, and 2b². Each of these multiplications will then be summed together to yield the final simplified expression. This process not only simplifies the expression but also provides valuable insights into how terms interact with each other in algebraic manipulations. Understanding the mechanics of this process is key to mastering algebraic simplification and problem-solving. By the end of this exercise, you will not only be able to solve this specific problem but also gain a deeper appreciation for the power and versatility of the distributive property in various mathematical contexts. The subsequent sections will guide you through each step, ensuring clarity and understanding at every stage of the process.
Step-by-Step Solution
Step 1: Applying the Distributive Property
The first step in multiplying 5ab³ by (4a² + 3ab + 2b²) is to apply the distributive property. This involves multiplying 5ab³ by each term inside the parentheses individually. So, we have: 5ab³ * 4a², 5ab³ * 3ab, and 5ab³ * 2b². This breaks down the original expression into three separate multiplication problems, each of which is more manageable to solve. The distributive property allows us to transform a single complex multiplication into a series of simpler multiplications, which is a fundamental strategy in algebraic manipulation. By distributing 5ab³ across the trinomial, we set the stage for simplifying each term by combining like factors. This initial step is crucial for a successful application of the distributive property, as it lays the groundwork for the subsequent steps of simplification and combination of terms. Understanding this process is vital for mastering algebraic expressions and equations, as it provides a structured approach to handling complex multiplications. In essence, we are spreading the multiplication over the addition, which is the core principle behind the distributive property. The next steps will focus on performing each of these multiplications and then combining the resulting terms.
Step 2: Multiplying Individual Terms
Now that we've distributed 5ab³ across the trinomial, we proceed to multiply each pair of terms. First, let's multiply 5ab³ by 4a². This involves multiplying the coefficients (5 and 4) and then multiplying the variables with the same base (a and a², and b³). When multiplying variables with exponents, we add the exponents. So, 5ab³ * 4a² = (5 * 4) * (a¹ * a²) * b³ = 20a³b³. Next, we multiply 5ab³ by 3ab. Again, we multiply the coefficients (5 and 3) and the variables with the same base (a and a, and b³ and b). Thus, 5ab³ * 3ab = (5 * 3) * (a¹ * a¹) * (b³ * b¹) = 15a²b⁴. Finally, we multiply 5ab³ by 2b². We multiply the coefficients (5 and 2) and the variables with the same base (b³ and b²). This gives us 5ab³ * 2b² = (5 * 2) * a¹ * (b³ * b²) = 10ab⁵. Each of these multiplications has resulted in a simplified term, which we will combine in the next step. This process of multiplying individual terms is a direct application of the rules of exponents and coefficients in algebra. By breaking down the multiplication into smaller, more manageable steps, we reduce the complexity and increase the accuracy of our calculations. The results of these individual multiplications will form the basis for our final simplified expression.
Step 3: Combining the Results
The final step is to combine the results from the previous multiplications. We have 20a³b³, 15a²b⁴, and 10ab⁵. Since these terms are not like terms (they have different combinations of exponents for the variables a and b), we cannot simplify them further by adding their coefficients. Therefore, the final expression is the sum of these terms: 20a³b³ + 15a²b⁴ + 10ab⁵. This expression represents the simplified form of the original expression 5ab³(4a² + 3ab + 2b²). The process of combining the results highlights the importance of identifying like terms in algebraic expressions. Like terms have the same variables raised to the same powers, allowing us to combine their coefficients. In this case, the absence of like terms means that the expression is already in its simplest form. This result underscores the efficiency of the distributive property in expanding and simplifying algebraic expressions. By breaking down the complex multiplication into smaller, manageable steps and then combining the results, we have successfully applied the distributive property to simplify the original expression. The final expression, 20a³b³ + 15a²b⁴ + 10ab⁵, is the expanded and simplified form of the given problem.
Final Answer
Therefore, after applying the distributive property and simplifying, the result of multiplying 5ab³ by (4a² + 3ab + 2b²) is 20a³b³ + 15a²b⁴ + 10ab⁵. This final answer represents the expanded form of the original expression, where the monomial 5ab³ has been successfully distributed across the terms of the trinomial (4a² + 3ab + 2b²). The process involved several key steps: first, distributing the monomial across each term of the trinomial; second, performing the individual multiplications, paying close attention to the rules of exponents and coefficients; and third, combining the resulting terms. In this case, the resulting terms were not like terms, so they could not be further simplified. The final answer, 20a³b³ + 15a²b⁴ + 10ab⁵, is a clear and concise representation of the expanded expression. This exercise demonstrates the power and utility of the distributive property in simplifying algebraic expressions and forms a fundamental skill for success in algebra and beyond. Understanding and mastering this property is crucial for tackling more complex mathematical problems and is a cornerstone of algebraic manipulation. The solution presented here provides a step-by-step guide to applying the distributive property, making it easier to understand and apply in future scenarios.
Conclusion
In conclusion, the distributive property is a powerful tool in mathematics that allows us to simplify expressions involving multiplication and addition. By multiplying 5ab³ by (4a² + 3ab + 2b²), we have demonstrated a practical application of this property. The step-by-step solution provided a clear pathway to expanding and simplifying algebraic expressions, highlighting the importance of distributing the monomial, multiplying individual terms, and combining like terms. The final result, 20a³b³ + 15a²b⁴ + 10ab⁵, showcases the simplified form of the original expression. Mastering the distributive property is not just about solving specific problems; it's about building a strong foundation in algebra and developing problem-solving skills that can be applied to a wide range of mathematical challenges. This property is a cornerstone of algebraic manipulation and is essential for success in more advanced mathematical topics. The ability to confidently apply the distributive property opens doors to simplifying complex equations, solving for unknowns, and understanding the relationships between different mathematical expressions. By understanding and practicing the distributive property, you are equipping yourself with a valuable tool that will serve you well throughout your mathematical journey. This article has aimed to provide a comprehensive understanding of the distributive property and its application, empowering you to tackle similar problems with confidence and precision.
