Simplifying (10+√3)(6+√2) A Step-by-Step Guide

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Introduction

In the realm of mathematics, simplifying expressions involving radicals can often seem daunting. However, by breaking down the process into manageable steps, these problems become significantly more approachable. This guide focuses on simplifying the expression (10+√3)(6+√2), providing a comprehensive, step-by-step approach that will not only help you solve this specific problem but also equip you with the skills to tackle similar expressions confidently. Our goal is to demystify the process, ensuring that every step is clear and logical. We will cover essential concepts such as the distributive property, multiplying radicals, and simplifying the resulting terms. By the end of this guide, you'll have a solid understanding of how to handle expressions of this nature and a sense of accomplishment in mastering this mathematical skill. So, let's embark on this mathematical journey together, unraveling the intricacies of simplifying radical expressions and transforming them into simpler, more understandable forms.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra, and it plays a crucial role in simplifying expressions like (10+√3)(6+√2). At its core, the distributive property allows us to multiply a single term by a group of terms within parentheses. In simpler terms, it states that for any numbers a, b, and c, a(b + c) = ab + ac. This seemingly simple rule is the key to expanding and simplifying more complex expressions. When we encounter an expression like (10+√3)(6+√2), we essentially need to apply the distributive property twice, a process often referred to as the FOIL method (First, Outer, Inner, Last). This involves multiplying each term in the first set of parentheses by each term in the second set. Breaking it down, we first multiply the "First" terms (10 and 6), then the "Outer" terms (10 and √2), followed by the "Inner" terms (√3 and 6), and finally the "Last" terms (√3 and √2). Understanding this process thoroughly is crucial as it forms the basis for the subsequent steps in simplifying the given expression. By mastering the distributive property, you'll be well-equipped to expand a wide range of algebraic expressions, making the simplification process much more manageable and less intimidating. This principle is not just confined to numerical expressions; it extends to algebraic expressions with variables and other mathematical constructs, making it a cornerstone of algebraic manipulation.

Step 1: Applying the Distributive Property (FOIL Method)

To begin simplifying the expression (10+√3)(6+√2), we employ the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that we multiply each term in the first binomial by each term in the second binomial. Let's break down each step:

  1. First: Multiply the first terms in each binomial: 10 * 6 = 60
  2. Outer: Multiply the outer terms: 10 * √2 = 10√2
  3. Inner: Multiply the inner terms: √3 * 6 = 6√3
  4. Last: Multiply the last terms in each binomial: √3 * √2 = √6

By systematically applying the FOIL method, we transform the original expression into a sum of these individual products. This step is crucial as it expands the expression, setting the stage for further simplification. The result of this application is: 60 + 10√2 + 6√3 + √6. This expanded form allows us to see all the terms that need to be considered and simplified. It's a critical transition from the initial product of binomials to a more manageable form where we can combine like terms and simplify radicals. Understanding this expansion process is essential for mastering algebraic manipulations and solving more complex mathematical problems. Therefore, a thorough grasp of the distributive property, as exemplified by the FOIL method, is a cornerstone of mathematical proficiency.

Step 2: Combining Like Terms

After applying the distributive property, our expression now looks like this: 60 + 10√2 + 6√3 + √6. The next crucial step in simplifying this expression is to identify and combine any like terms. In this context, like terms are those that have the same radical part. Looking at our expanded expression, we have four terms: 60, 10√2, 6√3, and √6. The term 60 is a constant, while the other terms involve square roots. To determine if any terms can be combined, we need to examine the radicals. We have √2, √3, and √6. Since these radicals are different (i.e., they do not contain the same number under the square root), none of the terms with radicals can be combined with each other. The constant term, 60, also cannot be combined with any of the radical terms because it lacks a radical component. Therefore, in this particular expression, there are no like terms that can be combined. This might seem like a negative result, but it's an important observation. Recognizing when terms cannot be combined is just as crucial as knowing when they can be. This step highlights the importance of careful observation and understanding the properties of radicals. In other expressions, you might find terms with the same radical (e.g., 2√5 and 7√5), which can be combined by adding their coefficients (in this case, 2 + 7 = 9, resulting in 9√5). However, in our current expression, the absence of like terms means we proceed to the next step without further simplification at this stage.

Step 3: Simplifying Radicals (If Possible)

Now that we have expanded our expression to 60 + 10√2 + 6√3 + √6 and confirmed that there are no like terms to combine, the next step is to examine each radical term to see if it can be simplified further. Simplifying radicals involves expressing the number under the square root (the radicand) as a product of its prime factors. If any of these prime factors appear in pairs, we can take one factor from each pair out of the square root. Let's consider each radical term individually:

  • √2: The number 2 is a prime number, meaning its only factors are 1 and itself. Therefore, √2 cannot be simplified any further.
  • √3: Similarly, 3 is a prime number, so √3 is already in its simplest form.
  • √6: The number 6 can be factored into 2 * 3. Since there are no pairs of identical factors, √6 also cannot be simplified further.

In this specific expression, none of the radicals can be simplified. This is not always the case, however. For instance, if we had a term like √8, we could simplify it because 8 can be factored into 2 * 2 * 2. This means we have a pair of 2s, so we can take a 2 out of the square root, leaving us with 2√2. The ability to simplify radicals is crucial because it allows us to express numbers in their simplest form, which can be particularly important in more complex calculations and mathematical proofs. Recognizing when a radical can be simplified and executing the simplification correctly is a key skill in algebra. In our case, since none of the radicals can be simplified, we proceed to the final step, which is to present the simplified expression.

Final Result

After meticulously applying the distributive property, combining like terms (which in this case, there were none), and attempting to simplify the radicals, we arrive at our final simplified expression. The steps we've taken ensure that we've explored all possible avenues for simplification, leaving us with the most concise form of the expression. Given our initial expression (10+√3)(6+√2), and after performing the necessary operations, the simplified form is:

60 + 10√2 + 6√3 + √6

This expression represents the culmination of our efforts. Each term has been carefully considered, and we've ensured that no further simplification is possible. The constant term, 60, stands alone, while the radical terms, 10√2, 6√3, and √6, each remain distinct due to the different numbers under the square root. This final result underscores the importance of a systematic approach to simplifying expressions. By breaking down the problem into smaller, manageable steps, we've been able to navigate the complexities of the expression and arrive at a clear and simplified answer. This process not only provides the solution but also reinforces the understanding of fundamental algebraic principles, such as the distributive property and the simplification of radicals. The final result, 60 + 10√2 + 6√3 + √6, is not just an answer; it's a testament to the power of methodical problem-solving in mathematics.

Conclusion

In summary, simplifying expressions involving radicals requires a systematic approach grounded in fundamental algebraic principles. In this guide, we've walked through the process of simplifying (10+√3)(6+√2), highlighting the importance of the distributive property, combining like terms, and simplifying radicals. By applying the FOIL method, we expanded the expression, and we carefully examined the resulting terms for any potential simplifications. We found that while there were no like terms to combine and the radicals were already in their simplest forms, the process underscored the necessity of these steps in any simplification problem. The final result, 60 + 10√2 + 6√3 + √6, represents the simplified form of the original expression. This exercise demonstrates that even expressions that may initially appear complex can be tackled effectively by breaking them down into smaller, more manageable steps. The key takeaways from this guide include a thorough understanding of the distributive property, the ability to identify and combine like terms, and the skill to simplify radicals. These are essential tools in algebra, and mastering them will enable you to confidently approach a wide range of mathematical problems. Simplifying expressions is not just about finding the right answer; it's about developing a logical and methodical approach to problem-solving, a skill that is valuable not only in mathematics but also in various aspects of life.