Simplifying $3ab(2ac + 3bc)$ A Step-by-Step Solution

In this comprehensive guide, we will meticulously dissect the algebraic expression $3ab(2ac + 3bc)$, guiding you through each step to arrive at the correct simplified form. Our goal is to not only provide the answer but also to foster a deep understanding of the underlying principles of algebraic manipulation. This exploration will be invaluable for students, educators, and anyone looking to reinforce their mathematical skills. Understanding the step-by-step process of simplifying algebraic expressions is a fundamental skill in mathematics. It forms the bedrock for more advanced topics such as calculus, linear algebra, and differential equations. By mastering these foundational concepts, you unlock the ability to tackle complex problems with confidence and precision. This article aims to provide a detailed walkthrough of the expression $3ab(2ac + 3bc)$, ensuring that readers not only grasp the solution but also the reasoning behind each step. We will break down the problem into manageable parts, making it accessible to learners of all levels. The journey through this algebraic expression will highlight the importance of the distributive property, the rules of exponents, and the art of combining like terms. By the end of this guide, you will be well-equipped to handle similar algebraic challenges and will have gained a deeper appreciation for the elegance of mathematical simplification. Algebraic expressions are the language of mathematics, and the ability to manipulate them is akin to fluency in this language. The expression $3ab(2ac + 3bc)$ is a prime example of how seemingly complex expressions can be simplified using fundamental algebraic principles. By working through this example, we aim to demystify algebraic simplification and empower readers to approach such problems with clarity and confidence. This guide will serve as a valuable resource for students, educators, and anyone seeking to strengthen their mathematical foundation. Let's embark on this mathematical journey together, unraveling the intricacies of algebraic simplification and unlocking the beauty of mathematical solutions. Mastering these concepts opens doors to a world of mathematical possibilities, enabling you to solve real-world problems and appreciate the elegance of mathematical reasoning. So, let's dive in and explore the intricacies of $3ab(2ac + 3bc) together, transforming complexity into clarity and building a solid foundation for mathematical success.

Step-by-Step Solution: Simplifying $3ab(2ac + 3bc)$

Let's embark on a detailed, step-by-step journey to simplify the given algebraic expression: $3ab(2ac + 3bc)$. Our primary tool in this simplification process will be the distributive property, a cornerstone of algebra. The distributive property dictates that for any numbers or algebraic terms a, b, and c, the following holds true: a( b + c ) = ab + ac. This seemingly simple rule is the key to unlocking the simplification of our expression. It allows us to break down the complex expression into smaller, more manageable parts. We will apply this property diligently, ensuring that each term within the parentheses is correctly multiplied by the term outside the parentheses. This meticulous approach is crucial to avoid errors and arrive at the accurate simplified form. Each step will be carefully explained, providing a clear understanding of the underlying logic and rationale. By mastering the distributive property, you gain a powerful tool for manipulating algebraic expressions and solving a wide range of mathematical problems. The distributive property is not merely a mathematical rule; it is a fundamental principle that reflects the way we distribute quantities in real-world scenarios. For instance, if you are buying 3 items each costing $5 + $2, you can calculate the total cost by multiplying 3 by the sum (5+2), or by multiplying 3 by each term separately (35 + 32), and then adding the results. The distributive property ensures that both methods yield the same answer. In the context of algebraic expressions, this property allows us to expand expressions, remove parentheses, and ultimately simplify them. The step-by-step application of the distributive property is like unfolding a complex map, revealing the hidden pathways and connections. Each step brings us closer to the final destination – the simplified form of the expression. So, let's begin our journey, armed with the distributive property, and unravel the intricacies of $3ab(2ac + 3bc) together. This journey will not only provide the solution but also equip you with a powerful tool for tackling future algebraic challenges. The distributive property is your ally in the world of algebraic simplification, and by mastering it, you empower yourself to navigate complex expressions with confidence and precision.

Applying the Distributive Property

The first crucial step in simplifying the expression $3ab(2ac + 3bc)$ is to apply the distributive property. This property, a fundamental concept in algebra, allows us to multiply a single term by a group of terms enclosed within parentheses. In our case, the term $3ab$ needs to be distributed across both $2ac$ and $3bc$. This means we will multiply $3ab$ by $2ac$ and then multiply $3ab$ by $3bc$. Each of these multiplications will result in a new term, and these terms will then be added together. The distributive property is not just a mathematical rule; it's a way of ensuring fairness in multiplication. It guarantees that each term inside the parentheses receives its due share of the multiplication. This property is a cornerstone of algebraic manipulation, and its mastery is essential for simplifying complex expressions and solving equations. Understanding the distributive property is like learning the alphabet of algebra; it's a foundational skill that unlocks the ability to read and write in the language of mathematics. The distributive property is a versatile tool that can be applied in a wide range of scenarios, from simple arithmetic calculations to complex algebraic equations. Its power lies in its ability to break down complex problems into smaller, more manageable parts. By applying the distributive property, we can transform expressions that appear daunting into simpler forms that are easier to understand and manipulate. This transformation is a key step in solving equations, simplifying expressions, and ultimately, gaining a deeper understanding of mathematical relationships. The application of the distributive property is like peeling away the layers of an onion, revealing the core essence of the expression. Each layer peeled away brings us closer to the simplified form, exposing the underlying structure and relationships. So, let's carefully apply the distributive property to $3ab(2ac + 3bc)$, ensuring that each term is multiplied correctly and that no errors are introduced. This meticulous approach will pave the way for a smooth and accurate simplification process. The distributive property is your guide, and with careful application, we will successfully navigate this algebraic terrain and arrive at the simplified form.

Performing the Multiplication: $3ab * 2ac$

Now, let's focus on the first multiplication: $3ab * 2ac$. When multiplying algebraic terms, we multiply the coefficients (the numerical parts) and then multiply the variables (the letter parts). In this case, the coefficients are 3 and 2, and the variables are a, b, and c. Multiplying the coefficients, we get 3 * 2 = 6. Next, we multiply the variables. Recall that when multiplying variables with the same base, we add their exponents. Here, we have a multiplied by a, which can be written as a¹ * a¹. Adding the exponents, we get a¹⁺¹ = a². The variable b appears only once, and the variable c also appears only once. Therefore, the product of the variables is a² * b * c, which is commonly written as a²bc. Combining the product of the coefficients and the product of the variables, we get 6a²bc. This resulting term, 6a²bc, is the first part of our simplified expression. The process of multiplying algebraic terms is like assembling a puzzle, where each piece (coefficient and variable) needs to be carefully combined to form the complete picture. The rules of exponents are the glue that holds these pieces together, ensuring that the variables are multiplied correctly. Mastery of these rules is essential for accurate algebraic manipulation. Multiplying algebraic terms is not just about following a set of rules; it's about understanding the underlying principles of mathematical operations. Each step, from multiplying the coefficients to adding the exponents, has a logical basis. By understanding these principles, you can approach algebraic multiplication with confidence and accuracy. The multiplication of $3ab$ and $2ac$ is a microcosm of the broader process of algebraic simplification. It showcases the importance of paying attention to detail, applying the rules of exponents correctly, and combining the results in a logical manner. This meticulous approach is the key to success in algebra. So, we have successfully navigated the first multiplication, arriving at the term 6a²bc. This term is a building block in our quest to simplify the original expression. Let's now move on to the second multiplication, applying the same principles and techniques to unlock the next piece of the puzzle. Each multiplication brings us closer to the final solution, and with careful execution, we will arrive at the simplified form with confidence.

Performing the Multiplication: $3ab * 3bc$

Now, let's tackle the second multiplication: $3ab * 3bc$. Following the same principles as before, we'll multiply the coefficients and then multiply the variables. The coefficients are 3 and 3, and their product is 3 * 3 = 9. For the variables, we have a, b, b, and c. The variable a appears only once. The variable b appears twice, which can be written as b¹ * b¹. Adding the exponents, we get b¹⁺¹ = b². The variable c appears only once. Therefore, the product of the variables is a * b² * c, commonly written as ab²c. Combining the product of the coefficients and the product of the variables, we get 9ab²c. This term, 9ab²c, is the second part of our simplified expression. The multiplication of $3ab$ and $3bc$ further reinforces the importance of understanding the rules of exponents and the process of multiplying algebraic terms. Each variable must be carefully considered, and its exponent must be adjusted accordingly. This meticulous approach ensures accuracy and avoids errors. This multiplication is like adding another piece to the puzzle, bringing us closer to the complete picture of the simplified expression. The term 9ab²c complements the term 6a²bc, and together, they form the simplified form of the original expression. The process of multiplying algebraic terms is not just a mechanical exercise; it's an opportunity to deepen our understanding of mathematical relationships. The variables and their exponents represent quantities and their interactions, and by multiplying them correctly, we gain insights into these relationships. So, we have successfully completed the second multiplication, arriving at the term 9ab²c. This term, combined with the previous term, will give us the final simplified expression. Let's now put the pieces together and see the result of our efforts. Each step has been carefully executed, and we are now on the verge of reaching our goal. The journey through this algebraic expression has been a testament to the power of the distributive property and the importance of meticulous calculation.

Combining the Results

Having performed both multiplications, we now have two terms: 6a²bc and 9ab²c. The final step is to combine these terms to obtain the simplified expression. Since these terms are not like terms (they have different variable combinations), we cannot combine them further. Therefore, the simplified expression is simply the sum of these two terms: 6a²bc + 9ab²c. This expression represents the simplified form of the original expression, $3ab(2ac + 3bc)$. We have successfully navigated the algebraic terrain, applying the distributive property and the rules of exponents to arrive at the final result. Combining the results is like completing the puzzle, where the individual pieces fit together to form the complete picture. The terms 6a²bc and 9ab²c are the final pieces, and their sum represents the simplified expression. The fact that these terms cannot be combined further highlights the importance of identifying like terms in algebraic expressions. Like terms have the same variable combinations, and only they can be added or subtracted. This distinction is crucial for accurate simplification. The simplified expression, 6a²bc + 9ab²c, is a testament to the power of algebraic manipulation. It demonstrates how a seemingly complex expression can be transformed into a simpler form through the application of fundamental principles. This simplification is not just an aesthetic improvement; it also makes the expression easier to understand and work with. So, we have reached the end of our journey, arriving at the simplified expression 6a²bc + 9ab²c. This expression is the culmination of our efforts, a symbol of our understanding of algebraic principles. The distributive property, the rules of exponents, and the process of combining like terms have all played their part in this successful simplification. Let's celebrate this achievement and carry the lessons learned forward, applying them to future algebraic challenges.

The Final Answer and its Significance

The final simplified form of the expression $3ab(2ac + 3bc)$ is 6a²bc + 9ab²c. This result is not just a collection of symbols; it's a concise representation of the original expression, revealing its underlying structure and relationships. The simplified form is easier to understand, manipulate, and use in further calculations. It's like having a clear roadmap instead of a tangled mess of lines. The significance of this simplification lies in its ability to transform complexity into clarity. The original expression, while mathematically sound, can be cumbersome to work with. The simplified form, on the other hand, is elegant and efficient. It allows us to see the essence of the expression without being bogged down by unnecessary details. This transformation is a key goal in mathematics: to express ideas and relationships in the simplest and most understandable way possible. The simplified expression 6a²bc + 9ab²c is a versatile tool that can be used in a variety of contexts. It can be substituted into equations, used to model real-world phenomena, or serve as a building block for more complex mathematical expressions. Its simplicity makes it accessible and adaptable, enhancing its value. The journey to arrive at this final answer has been a valuable learning experience. We have reinforced our understanding of the distributive property, the rules of exponents, and the process of combining like terms. These skills are not just applicable to this specific problem; they are transferable to a wide range of mathematical challenges. The final answer is not just the end of the process; it's a stepping stone to further mathematical exploration. It's an invitation to apply our knowledge to new problems, to deepen our understanding, and to appreciate the beauty and power of mathematics. So, let's celebrate this achievement and carry the lessons learned forward, embracing the challenges and opportunities that lie ahead. The simplified expression 6a²bc + 9ab²c is a symbol of our mathematical journey, a testament to our perseverance and our understanding.

Identifying the Correct Option

Now that we have meticulously simplified the expression $3ab(2ac + 3bc)$ to 6a²bc + 9ab²c, let's examine the provided options to identify the correct answer. The options presented are:

• A. $6abc + 3bc$
• B. $6a^2bc + 3bc$
• C. $6a^2bc + 9b^2c$
• D. $6a^2bc + 9ab^2c$

By carefully comparing our simplified expression, 6a²bc + 9ab²c, with the options, we can clearly see that option D, $6a^2bc + 9ab^2c$, matches our result. The other options contain terms that are either missing variables, have incorrect exponents, or both. This comparison underscores the importance of accurate simplification. A single mistake in the process can lead to an incorrect final answer. The meticulous step-by-step approach we employed ensured that we arrived at the correct result and could confidently identify the matching option. Identifying the correct option is like finding the missing piece of a puzzle, completing the picture and validating our efforts. The simplified expression serves as a key, unlocking the correct answer from a set of possibilities. This process of comparison and validation is a crucial part of problem-solving. It allows us to check our work, identify any errors, and ensure that our answer is consistent with the given information. The ability to identify the correct option is not just about finding the right answer; it's about demonstrating a deep understanding of the problem and the solution process. It's a testament to our ability to translate mathematical expressions, manipulate them accurately, and interpret the results correctly. So, with confidence, we can declare that option D is the correct answer, a validation of our careful and methodical approach. This success reinforces the importance of accuracy, attention to detail, and a thorough understanding of algebraic principles. Let's carry this confidence and these skills forward, applying them to future mathematical challenges.

Conclusion: Mastering Algebraic Simplification

In conclusion, we have successfully simplified the algebraic expression $3ab(2ac + 3bc)$ to 6a²bc + 9ab²c, identifying option D as the correct answer. This journey through algebraic manipulation has highlighted the importance of several key concepts and skills. We have reinforced our understanding of the distributive property, a cornerstone of algebraic simplification. We have applied the rules of exponents to multiply variables accurately. We have learned the significance of combining like terms and recognizing when terms cannot be combined further. And we have emphasized the value of a step-by-step, meticulous approach to problem-solving. Mastering algebraic simplification is not just about arriving at the correct answer; it's about developing a deep understanding of mathematical principles and building confidence in our ability to solve problems. The skills we have honed in this exercise are transferable to a wide range of mathematical contexts, from solving equations to graphing functions to modeling real-world phenomena. Algebraic simplification is a gateway to more advanced mathematical topics. It's a foundational skill that unlocks the ability to tackle complex problems with clarity and precision. By mastering these fundamental concepts, we empower ourselves to explore the vast and fascinating world of mathematics. The journey through algebraic simplification is a journey of discovery, where each step reveals new insights and reinforces our understanding. The expression $3ab(2ac + 3bc)$ served as our vehicle, guiding us through the landscape of algebraic principles. And with each step, we gained a deeper appreciation for the elegance and power of mathematics. So, let's celebrate our success and carry the lessons learned forward, embracing the challenges and opportunities that lie ahead. The mastery of algebraic simplification is a valuable asset, a key to unlocking mathematical potential and a foundation for lifelong learning. Let's continue to explore, to learn, and to grow in our mathematical journey.