Simplifying (9a²b² × 2b³a) / (6a⁴b²) A Step-by-Step Algebraic Guide

In the realm of algebra, simplifying expressions is a fundamental skill. It allows us to represent complex mathematical relationships in a concise and manageable form. This article will delve into the process of simplifying a specific algebraic expression: (9a²b² × 2b³a) / (6a⁴b²). We will break down each step, providing a clear and comprehensive understanding of the underlying principles. Algebraic expressions are the backbone of mathematical problem-solving, and the ability to simplify them is crucial for success in various mathematical disciplines. This guide aims to equip you with the necessary tools and techniques to tackle such problems confidently. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this article will serve as a valuable resource. We will explore the core concepts of exponents, coefficients, and the order of operations, all while maintaining a focus on clarity and simplicity. By the end of this guide, you will be able to not only simplify this particular expression but also apply the same principles to a wide range of algebraic problems. Let's embark on this journey of mathematical simplification together, unraveling the complexities of algebraic expressions and mastering the art of simplifying them.

Understanding the Basics of Algebraic Expressions

Before we dive into the simplification process, let's establish a solid understanding of the basic components of algebraic expressions. Algebraic expressions are combinations of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. A variable is a symbol (usually a letter) that represents an unknown value. In our expression, 'a' and 'b' are variables. A constant is a fixed numerical value, such as 9, 2, and 6 in our expression. The coefficients are the numerical factors that multiply the variables. For example, in the term 9a²b², 9 is the coefficient. The exponents indicate the power to which a variable is raised. In the term a², the exponent is 2, meaning 'a' is multiplied by itself. To effectively simplify algebraic expressions, we need to be comfortable with these components and how they interact with each other. Understanding the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is also crucial. This order dictates the sequence in which operations should be performed to arrive at the correct answer. In the context of simplifying expressions, we often deal with terms that involve variables raised to different powers. The rules of exponents play a vital role in these scenarios. For instance, when multiplying terms with the same base, we add the exponents. When dividing terms with the same base, we subtract the exponents. These rules will be instrumental in simplifying our expression. By mastering these foundational concepts, we set the stage for a smooth and successful simplification process. Let's now move on to the specific steps involved in simplifying the given expression.

Step 1: Rewriting the Expression

The initial expression we need to simplify is (9a²b² × 2b³a) / (6a⁴b²). The first step in simplifying any complex expression is to rewrite it in a more organized manner. This involves expanding the multiplication in the numerator and expressing the division as a fraction. By rewriting the expression, we make it easier to identify the different components and apply the relevant simplification rules. The original expression involves multiplication and division of terms containing variables and constants. By separating these operations, we can tackle them individually, which simplifies the overall process. This step also helps in visualizing the structure of the expression, making it easier to identify common factors and apply the rules of exponents. Rewriting the expression is not just about making it look neater; it's about transforming it into a format that is more conducive to simplification. It's a strategic move that sets the stage for the subsequent steps. When rewriting, we pay close attention to the order of operations and ensure that the expression remains mathematically equivalent to the original. This means that the value of the expression should not change as a result of the rewriting process. For example, we can rewrite the multiplication in the numerator as a single term, combining the constants and variables. This step prepares us for the next stage, where we will apply the rules of exponents to simplify the variable terms. By the end of this step, we will have a clearer picture of the expression's structure, which will make the simplification process more straightforward. Now, let's proceed to the next step, where we will apply the rules of exponents to simplify the expression further.

Step 2: Simplifying the Numerator

Now that we have rewritten the expression, our next focus is on simplifying the numerator. The numerator of our expression is 9a²b² × 2b³a. To simplify this, we need to multiply the coefficients and combine the variables with the same base by adding their exponents. This step involves applying the fundamental rules of exponents, which are crucial for simplifying algebraic expressions. The product rule of exponents states that when multiplying terms with the same base, we add the exponents. For example, a² × a = a^(2+1) = a³. Similarly, b² × b³ = b^(2+3) = b⁵. By applying this rule, we can combine the 'a' terms and the 'b' terms in the numerator. The coefficients, 9 and 2, are simply multiplied together. This step is a prime example of how algebraic simplification can transform a complex expression into a simpler one. It requires a careful application of the rules of exponents and a methodical approach to ensure accuracy. Simplifying the numerator not only makes the expression easier to handle but also reveals the underlying structure of the expression. It helps in identifying common factors that can be canceled out in the later steps. This step is also an excellent opportunity to reinforce the understanding of exponent rules and their application in simplifying algebraic expressions. By breaking down the numerator into its simplest form, we pave the way for the next step, where we will simplify the entire expression by considering the denominator. Let's move on to the next step and continue our journey of simplifying the given algebraic expression.

Step 3: Simplifying the Entire Expression

With the numerator simplified, we can now focus on simplifying the entire expression. We have the expression 18a³b⁵ / 6a⁴b². This step involves dividing the coefficients and applying the quotient rule of exponents. The quotient rule states that when dividing terms with the same base, we subtract the exponents. This rule is the counterpart to the product rule and is equally essential in simplifying algebraic expressions. For example, a³ / a⁴ = a^(3-4) = a⁻¹. Similarly, b⁵ / b² = b^(5-2) = b³. In this step, we also divide the coefficients, 18 and 6, which is a straightforward arithmetic operation. Simplifying the entire expression involves a combination of arithmetic and the application of exponent rules. It requires careful attention to detail and a methodical approach to ensure accuracy. This step is where all the previous steps come together, and the expression is reduced to its simplest form. The simplified expression will be mathematically equivalent to the original but will be in a more concise and manageable form. This step is also an excellent opportunity to reinforce the understanding of the quotient rule of exponents and its application in simplifying algebraic expressions. By the end of this step, we will have a fully simplified expression, which represents the final answer. Let's move on to the final step, where we will express the result in its simplest form and discuss the implications of the negative exponent.

Step 4: Expressing the Result in Simplest Form

After applying the quotient rule, we arrive at the expression 3a⁻¹b³. However, it's customary to express algebraic expressions with positive exponents. To do this, we use the rule that a⁻ⁿ = 1/aⁿ. Applying this rule to our expression, we can rewrite a⁻¹ as 1/a. This final step ensures that our answer is in the most conventional and simplified form. Expressing the result with positive exponents is not just a matter of convention; it also makes the expression easier to interpret and use in further calculations. A negative exponent indicates a reciprocal, and by rewriting it as a positive exponent in the denominator, we make the relationship between the variables and constants clearer. This step also highlights the importance of understanding the different rules of exponents and how they can be used to manipulate algebraic expressions. By rewriting the expression with positive exponents, we complete the simplification process and arrive at the final answer. This step is also a good opportunity to review all the steps we have taken and ensure that the final answer is mathematically equivalent to the original expression. Now, let's put it all together and present the final simplified form of the expression. The final simplified expression is 3b³/a. This is the most concise and manageable form of the original expression, and it represents the culmination of our efforts. We have successfully simplified the given algebraic expression by applying the rules of exponents and simplifying the coefficients. This process demonstrates the power of algebraic simplification in reducing complex expressions to their simplest forms. Now, let's move on to a brief conclusion to summarize what we have learned.

Conclusion

In this article, we have successfully simplified the algebraic expression (9a²b² × 2b³a) / (6a⁴b²) to its simplest form, which is 3b³/a. We achieved this by following a step-by-step approach, which included rewriting the expression, simplifying the numerator, simplifying the entire expression, and expressing the result with positive exponents. This process demonstrates the importance of understanding the fundamental rules of algebra, particularly the rules of exponents, in simplifying complex expressions. Simplifying algebraic expressions is a crucial skill in mathematics, as it allows us to represent mathematical relationships in a concise and manageable form. It is also a foundational skill for more advanced mathematical concepts. By mastering the techniques presented in this article, you will be well-equipped to tackle a wide range of algebraic problems. The ability to simplify expressions not only makes problem-solving easier but also enhances your understanding of the underlying mathematical principles. This article has provided a comprehensive guide to simplifying a specific algebraic expression, but the principles and techniques discussed can be applied to many other similar problems. The key is to approach each problem methodically, break it down into smaller steps, and apply the relevant rules and principles. With practice and a solid understanding of the fundamentals, you can become proficient in simplifying algebraic expressions and excel in your mathematical endeavors. We hope this guide has been helpful and has provided you with the necessary tools to confidently simplify algebraic expressions. Remember, mathematics is a journey of learning and discovery, and every problem solved is a step forward.