Simplifying (5√3 + √2)(√3 - √2) A Step By Step Guide

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Hey guys! Today, let's dive into simplifying the expression (5√3 + √2)(√3 - √2). This might look a bit intimidating at first glance, but don't worry, we're going to break it down step-by-step. We'll cover the basic principles of simplifying radical expressions, walk through the multiplication process, and then consolidate everything to get our final, simplified answer. So, grab your math hats, and let's get started!

Understanding the Basics of Radical Expressions

Before we jump into the main problem, let's quickly recap what radical expressions are and how they work. Radical expressions involve roots, like square roots (√), cube roots, and so on. The square root of a number x (√x) is a value that, when multiplied by itself, equals x. For instance, √9 = 3 because 3 * 3 = 9. When dealing with radical expressions, especially in multiplication, we need to remember a few key properties. One of the most important is the distributive property, which we'll use extensively in this problem. The distributive property states that a(b + c) = ab + ac. This principle allows us to multiply a term by a group of terms inside parentheses. Another important concept is how to multiply radicals. The rule is straightforward: √a * √b = √(a * b). This means we can multiply the numbers inside the square roots directly. For example, √2 * √3 = √6. Additionally, we'll need to remember how to simplify radicals. If a number inside the square root has a perfect square factor, we can simplify it further. For instance, √12 can be simplified because 12 has a perfect square factor of 4 (12 = 4 * 3). So, √12 = √(4 * 3) = √4 * √3 = 2√3. Keeping these basics in mind will make the entire process much smoother. Radicals might seem tricky at first, but with a bit of practice, you'll get the hang of them. Think of them as another type of number that follows specific rules, just like fractions or decimals. The key is to understand these rules and apply them systematically. Now that we've brushed up on the basics, let's tackle our main problem: simplifying (5√3 + √2)(√3 - √2).

Step-by-Step Multiplication of the Expression

Now, let's get to the heart of the matter: multiplying (5√3 + √2)(√3 - √2). To do this, we'll use the FOIL method, which is a handy way to remember how to apply the distributive property when multiplying two binomials (expressions with two terms). FOIL stands for: First, Outer, Inner, Last. This tells us the order in which to multiply the terms:

  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms in the expression.
  • Inner: Multiply the inner terms.
  • Last: Multiply the last terms in each binomial.

Let's apply this to our expression:

  1. First: (5√3) * (√3) = 5 * (√3 * √3) = 5 * 3 = 15
  2. Outer: (5√3) * (-√2) = -5 * (√3 * √2) = -5√6
  3. Inner: (√2) * (√3) = √(2 * 3) = √6
  4. Last: (√2) * (-√2) = -(√2 * √2) = -2

So, after applying the FOIL method, we have: 15 - 5√6 + √6 - 2. This is a crucial step, so make sure you take your time and double-check each multiplication. A small mistake here can throw off the whole solution. The FOIL method is a lifesaver for these types of problems, and it’s something you’ll use again and again in algebra. It’s all about being organized and methodical. Write out each step clearly, and you’ll minimize the chances of making errors. Remember, practice makes perfect! The more you use the FOIL method, the more comfortable you’ll become with it. Now that we’ve expanded our expression, the next step is to simplify it by combining like terms. This will bring us closer to our final answer.

Combining Like Terms and Simplifying

Alright, we've successfully multiplied the terms, and now we have the expression 15 - 5√6 + √6 - 2. The next step is to combine like terms to simplify this expression. Like terms are terms that have the same radical part. In our expression, we have two types of terms: integers (whole numbers) and terms with √6. The integers are 15 and -2, and the terms with √6 are -5√6 and √6. Let's combine the integers first: 15 - 2 = 13. Now, let's combine the terms with √6. Think of √6 as a variable, like x. So, -5√6 + √6 is similar to -5x + x. To combine these, we add their coefficients (the numbers in front of the radical). In this case, the coefficients are -5 and 1 (since √6 is the same as 1√6). So, -5 + 1 = -4. Therefore, -5√6 + √6 = -4√6. Now we can put it all together. We have 13 from combining the integers and -4√6 from combining the radical terms. This gives us our simplified expression: 13 - 4√6. This is our final answer! We've taken a somewhat complex expression and simplified it down to its most basic form. Combining like terms is a fundamental skill in algebra. It allows us to take long, complicated expressions and make them much easier to work with. Always remember to look for terms that have the same variable part (in this case, the radical part) and then combine their coefficients. This step is crucial for simplifying any algebraic expression, not just those involving radicals. So, make sure you're comfortable with this process. Now, let's recap the entire process to make sure we've got it all down.

Final Simplified Answer

Let's recap what we've done to simplify the expression (5√3 + √2)(√3 - √2). We started by understanding the basics of radical expressions and the distributive property. Then, we used the FOIL method to multiply the two binomials:

  1. First: (5√3)(√3) = 15
  2. Outer: (5√3)(-√2) = -5√6
  3. Inner: (√2)(√3) = √6
  4. Last: (√2)(-√2) = -2

This gave us the expression 15 - 5√6 + √6 - 2. Next, we combined like terms:

  • Combined the integers: 15 - 2 = 13
  • Combined the radical terms: -5√6 + √6 = -4√6

Finally, we put it all together to get our simplified answer: 13 - 4√6. So, there you have it! We've successfully simplified the expression. Remember, the key to simplifying these types of problems is to break them down into smaller, manageable steps. Understand the rules, apply them methodically, and don't rush the process. Math is like building a house – you need a solid foundation to build something strong. Each step we took is a crucial part of that foundation. Whether it's understanding the distributive property, applying the FOIL method, or combining like terms, each concept plays a vital role in simplifying expressions. Keep practicing, and you'll find these problems become easier and easier. And that’s the final answer, guys! I hope this breakdown has helped you understand the process a bit better. Feel free to try out similar problems to sharpen your skills. Happy simplifying!