Simplifying The Expression $3 \sqrt{12} + 3 \sqrt{3} - 4 \sqrt{2}$ A Step-by-Step Guide

Jul 14, 2025 by ADMIN 88 views

$Simplify the Expression: $3 \sqrt{12} + 3 \sqrt{3} - 4 \sqrt{2}$$

Introduction

In the realm of mathematics, simplifying expressions is a fundamental skill. Mathematical expressions often appear in complex forms, making them difficult to understand and work with. Simplification allows us to rewrite these expressions in a more manageable and transparent way. This article delves into the process of simplifying a specific mathematical expression: $3 \sqrt{12} + 3 \sqrt{3} - 4 \sqrt{2}$ . This expression involves square roots and requires a systematic approach to simplify effectively. Mastering the technique to simplify such expressions is critical for problem-solving in various areas of mathematics, including algebra, calculus, and geometry. We will explore the step-by-step method, focusing on how to break down the terms, combine like terms, and arrive at the simplest form. By understanding this process, readers will be better equipped to handle similar simplification problems and enhance their mathematical abilities. This discussion will not only cover the mechanics of simplification but also emphasize the importance of understanding the underlying principles, which is crucial for applying these skills in more complex scenarios.

Breaking Down the Expression

The first step in simplifying the expression $3 \sqrt{12} + 3 \sqrt{3} - 4 \sqrt{2}$ is to break down each term individually. We'll focus on simplifying the square root terms, as these are often the components that can be reduced. The key here is to identify perfect square factors within the radicands (the numbers inside the square root). Let’s start with $3 \sqrt{12}$ . The number 12 can be factored into $4 \times 3$ , where 4 is a perfect square ( $2^2$ ). So, we can rewrite $\sqrt{12}$ as $\sqrt{4 \times 3}$ . Next, we apply the property of square roots that states $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ . This allows us to separate the square root into $\sqrt{4} \times \sqrt{3}$ . Since $\sqrt{4}$ is 2, the expression becomes $2 \sqrt{3}$ . Multiplying this by the original coefficient 3, we get $3 \times 2 \sqrt{3}$ , which simplifies to $6 \sqrt{3}$ . Now, let's look at the term $3 \sqrt{3}$ . The number 3 is a prime number and has no perfect square factors other than 1, so $\sqrt{3}$ remains as it is. The term $3 \sqrt{3}$ is already in its simplest form. Finally, consider the term $-4 \sqrt{2}$ . Similarly, the number 2 is also a prime number, and $\sqrt{2}$ cannot be simplified further. Thus, the term $-4 \sqrt{2}$ remains unchanged. By breaking down each term, we have transformed the original expression into a form where we can identify and combine like terms. This initial step is crucial for simplifying more complex expressions and sets the stage for the subsequent steps in the simplification process.

Combining Like Terms

After breaking down the expression, the next crucial step is to combine like terms. In the simplified form, we have $6 \sqrt{3} + 3 \sqrt{3} - 4 \sqrt{2}$ . Like terms are terms that have the same radical part (the square root part). In this expression, $6 \sqrt{3}$ and $3 \sqrt{3}$ are like terms because they both contain $\sqrt{3}$ . The term $-4 \sqrt{2}$ is not a like term with the others because it contains $\sqrt{2}$ , which is different from $\sqrt{3}$ . To combine like terms, we simply add or subtract their coefficients (the numbers in front of the square roots). In this case, we add the coefficients of $6 \sqrt{3}$ and $3 \sqrt{3}$ , which are 6 and 3, respectively. Adding 6 and 3 gives us 9, so $6 \sqrt{3} + 3 \sqrt{3}$ simplifies to $9 \sqrt{3}$ . The term $-4 \sqrt{2}$ remains unchanged as there are no other terms with $\sqrt{2}$ to combine with. Therefore, the expression $6 \sqrt{3} + 3 \sqrt{3} - 4 \sqrt{2}$ simplifies to $9 \sqrt{3} - 4 \sqrt{2}$ . This step of combining like terms is vital because it reduces the number of terms in the expression and makes it easier to understand and use in further calculations. Recognizing and combining like terms is a fundamental skill in algebra and is essential for simplifying various types of algebraic expressions.

Final Simplified Expression

After breaking down the terms and combining like terms, we arrive at the final simplified expression. From our previous steps, we had $6 \sqrt{3} + 3 \sqrt{3} - 4 \sqrt{2}$ , which simplified to $9 \sqrt{3} - 4 \sqrt{2}$ . At this point, we have examined each term and ensured that the square roots are simplified as much as possible. We have also combined all like terms, leaving us with two terms that cannot be further combined because they contain different radicals ( $\sqrt{3}$ and $\sqrt{2}$ ). The expression $9 \sqrt{3} - 4 \sqrt{2}$ is now in its simplest form. There are no more perfect square factors to extract from the square roots, and there are no more like terms to combine. This final form is the most concise and understandable representation of the original expression. It is important to recognize when an expression is fully simplified, as this ensures that the answer is in its most manageable and useful form. The ability to simplify expressions to their final form is a critical skill in mathematics, allowing for easier manipulation and application in more complex problems. Understanding this process thoroughly will enhance one's mathematical proficiency and problem-solving abilities.

Conclusion

In conclusion, simplifying the expression $3 \sqrt{12} + 3 \sqrt{3} - 4 \sqrt{2}$ involves a systematic approach that includes breaking down terms, identifying like terms, and combining them. The initial step of breaking down $3 \sqrt{12}$ into $6 \sqrt{3}$ is crucial, as it reveals the like terms within the expression. Subsequently, combining the like terms $6 \sqrt{3}$ and $3 \sqrt{3}$ yields $9 \sqrt{3}$ . The term $-4 \sqrt{2}$ remains unchanged as it has no like terms to combine with. The final simplified expression is $9 \sqrt{3} - 4 \sqrt{2}$ , which cannot be simplified further. This process highlights the importance of recognizing perfect square factors within square roots and understanding the rules for combining like terms. Mastering these techniques is fundamental for simplifying more complex algebraic expressions and is a valuable skill in various mathematical contexts. By following a step-by-step approach, one can confidently simplify expressions and arrive at the most concise and understandable form. This not only aids in problem-solving but also enhances overall mathematical comprehension and proficiency. The ability to simplify expressions is a cornerstone of mathematical literacy and is essential for success in advanced mathematical studies.