Simplifying Radical Expressions: A Step-by-Step Guide

Hey guys! Let's dive into simplifying the expression: √2 * (√8 + 2√6) - √3 * (√27 + 3√6). This might look a bit intimidating at first, with all those square roots floating around, but trust me, it's just a matter of breaking things down step by step. We'll go through it together, so you can see how to tackle these kinds of problems. Understanding how to simplify radical expressions is a fundamental skill in algebra, and it's super useful for all sorts of math problems. So, grab your calculators (optional, but helpful!), and let's get started. We will break down the expression and go through each step carefully. By the end, you'll be a pro at simplifying these radical expressions. This skill is very useful for further maths and complex equation-solving.

Breaking Down the Expression

Okay, the expression we're dealing with is √2 * (√8 + 2√6) - √3 * (√27 + 3√6). The first thing we want to do is look at the square roots and see if we can simplify any of them. Remember, the goal is to get rid of square roots by finding perfect squares inside of them. Perfect squares are numbers like 4, 9, 16, 25, and so on – numbers that have whole number square roots. For example, the square root of 9 is 3. So, let's start with the term √8. We can rewrite √8 as √(4 * 2). Since 4 is a perfect square, we can simplify this to 2√2. Next, we have 2√6, which we can't simplify further because 6 doesn't have any perfect square factors other than 1. Moving on to the second part of our expression, let's simplify √27. We can rewrite √27 as √(9 * 3). Since 9 is a perfect square, we can simplify this to 3√3. Finally, we have 3√6, which again we can't simplify further.

So, our expression now looks like this: √2 * (2√2 + 2√6) - √3 * (3√3 + 3√6). It's already looking a bit cleaner, right? Remember, the key here is to find perfect squares and take their square roots out of the radical. This reduces the complexity and makes the entire equation easier to solve. Always be sure to check for the greatest perfect square factors to fully simplify the radical. This step-by-step process breaks the complex problems into smaller, manageable parts. Don't rush, it's important to take your time and be accurate while doing the steps.

Simplifying √8 and √27

As we discussed, √8 can be simplified to 2√2 because √8 = √(4 * 2) = √4 * √2 = 2√2. Similarly, √27 simplifies to 3√3 because √27 = √(9 * 3) = √9 * √3 = 3√3. This simplification is the cornerstone of making our original expression manageable.

Distributing and Further Simplification

Now that we've simplified the square roots within the parentheses, we need to distribute. This means we'll multiply the terms outside the parentheses by each term inside the parentheses. Let's do the first part of the expression: √2 * (2√2 + 2√6). Multiplying √2 by 2√2 gives us 2 * (√2 * √2), which simplifies to 2 * 2 = 4. Multiplying √2 by 2√6 gives us 2√12. Now, let's do the second part: √3 * (3√3 + 3√6). Multiplying √3 by 3√3 gives us 3 * (√3 * √3), which simplifies to 3 * 3 = 9. Multiplying √3 by 3√6 gives us 3√18. So, our expression now looks like this: 4 + 2√12 - 9 - 3√18. We're getting closer to the solution, and it's becoming more manageable with each step.

We've now got a bunch of terms, some with radicals and some without. Our next step is to simplify those radicals further if possible. Let's look at 2√12. We can rewrite √12 as √(4 * 3), which simplifies to 2√3. So, 2√12 becomes 2 * 2√3 = 4√3. Then, let's look at 3√18. We can rewrite √18 as √(9 * 2), which simplifies to 3√2. So, 3√18 becomes 3 * 3√2 = 9√2. Now we are going to combine the constants and simplified the equation. We are almost there, keep going, it will get easier. Make sure that you note all the steps, this process might seem hard at first but with a bit of practice it will be as easy as it can get.

Distributing the Terms

Distributing the terms is a critical step. For the first part: √2 * (2√2 + 2√6), we get 4 + 2√12. For the second part: √3 * (3√3 + 3√6), we get 9 + 3√18. This step prepares us to combine like terms and complete the calculation.

Final Simplification and Calculation

At this point, our expression is 4 + 4√3 - 9 - 9√2, combining the like terms, we can combine the constants which are 4 and -9. 4-9 = -5. Now we are left with -5 + 4√3 - 9√2. Now we can combine any other like terms. In this case, we don't have any like terms that we can combine. Our final expression looks like this -5 + 4√3 - 9√2. We might be tempted to think that this is the final answer, but, let's double-check everything to make sure. Since none of the terms can be simplified any further, that's it! Our simplified expression is -5 + 4√3 - 9√2. In this case, the answer is not a whole number, and it's not one of the answer options given. Let's revisit our initial steps to ensure we didn't miss any. The key is to simplify the square roots, distribute, and then combine the like terms. If the provided answer options do not fit, then it would be