Simplifying 6√√2 + 4√2 - 10√2 A Step-by-Step Guide

Hey guys! Let's break down this math problem together. We're going to simplify the expression 6√√2 + 4√2 - 10√2 step by step. Don't worry, it's not as scary as it looks! We'll go through each part, explain the process, and by the end, you'll be a pro at simplifying expressions like this one. So, grab your pencils and let's dive in!

Understanding the Basics of Radicals

Before we jump into simplifying our expression, it's super important to get a handle on what radicals actually are. Think of radicals as the opposite of exponents. The most common radical is the square root (√), which asks the question, “What number, when multiplied by itself, equals this number?” For example, √9 = 3 because 3 * 3 = 9.

Now, let's talk about the anatomy of a radical. The symbol ‘√’ is called the radical sign. The number inside the radical sign is called the radicand, and the small number perched on the radical sign (if there is one) is called the index. When you see a square root without an index, it's understood to be 2. So, √x is the same as ²√x. Other common radicals include cube roots (³√), fourth roots (⁴√), and so on.

The key concept here is understanding how radicals interact with each other and with regular numbers. You can only combine radicals that have the same index and radicand. For example, you can add 2√5 and 3√5 because they both have the same square root (√) and the same radicand (5). However, you can't directly add 2√5 and 3√7 because they have different radicands. Similarly, you can't directly add √5 and ³√5 because they have different indices.

When we're dealing with expressions involving radicals, it’s often necessary to simplify them first. This usually involves breaking down the radicand into its prime factors and looking for pairs (in the case of square roots), triplets (in the case of cube roots), and so on. For example, √20 can be simplified because 20 can be factored into 2 * 2 * 5. Since we have a pair of 2s, we can take one 2 out of the radical, leaving us with 2√5. This simplification makes it much easier to combine like terms and perform other operations.

Understanding these basic principles is crucial for tackling more complex problems, and it's the foundation for simplifying our expression 6√√2 + 4√2 - 10√2. So, let's move on and see how we can apply these concepts!

Breaking Down 6√√2

Okay, let's tackle the first term in our expression: 6√√2. This might look a little intimidating at first because we've got a radical inside another radical! But don't worry, we'll break it down nice and easy.

First things first, let's remember what a radical actually means. The square root of a number (√x) is the same as raising that number to the power of 1/2 (x^(1/2)). So, when we see √√2, we can think of it as the square root of the square root of 2. Mathematically, this can be written as √(√2) = (2^(1/2))(1/2).

Now, here's a cool trick: when you raise a power to another power, you multiply the exponents. So, (2^(1/2))(1/2) becomes 2^((1/2)*(1/2)) = 2^(1/4). This means √√2 is the same as the fourth root of 2 (⁴√2). See? We're already making progress!

So, let's rewrite our term 6√√2 using this new understanding. We now have 6 * 2^(1/4), or 6⁴√2. This form is much simpler to work with because we've eliminated the nested radicals. We've essentially transformed a complex expression into something more manageable by applying the rules of exponents and radicals.

Now that we've simplified 6√√2 to 6⁴√2, let's think about what this means in the context of our original problem. We've taken a potentially confusing term and made it clearer. This step is crucial because it allows us to see how this term relates to the other terms in the expression. Remember, the goal is to combine like terms, and to do that effectively, we need to express everything in its simplest form.

By breaking down 6√√2, we’ve not only simplified the term itself but also gained a better understanding of how radicals and exponents work together. This is a fundamental skill in algebra and will help you tackle many other problems involving radicals. So, keep this in mind as we move on to the next steps in simplifying the entire expression. Next up, we'll look at the other terms and see how we can combine them with our simplified term.

Dealing with 4√2 - 10√2

Alright, now let's shift our focus to the remaining terms in the expression: 4√2 - 10√2. These terms look much simpler than our earlier 6√√2, and that's great news! The key here is to recognize that these are like terms. Remember, like terms are terms that have the same variable (or in this case, the same radical) raised to the same power.

In our case, both terms have the square root of 2 (√2). This means we can combine them just like we would combine 4x and -10x. Think of √2 as a variable; it makes the process much easier to visualize. So, we have 4 “something” minus 10 “something,” where the “something” is √2.

To combine these terms, we simply add (or subtract) their coefficients. The coefficient is the number in front of the radical. In this case, we have 4 and -10. So, we perform the operation 4 - 10, which equals -6. Therefore, 4√2 - 10√2 simplifies to -6√2.

This simplification is a perfect example of why it's important to understand the basic rules of algebra. Combining like terms is a fundamental skill, and it's essential for simplifying expressions and solving equations. By recognizing the common radical (√2) in both terms, we were able to quickly and easily reduce two terms into one.

Now, let's take a moment to appreciate what we've accomplished. We've tackled the trickier term 6√√2 and simplified it to 6⁴√2. We've also combined 4√2 - 10√2 into -6√2. We're now in a position to bring all the pieces together and see if we can further simplify the entire expression. Remember, our goal is to make the expression as simple as possible, and we're well on our way to achieving that!

So, with these simplifications in hand, let's move on to the final step: putting everything together and seeing if we can simplify further. Are we able to combine 6⁴√2 with -6√2? Let's find out!

Putting It All Together: 6⁴√2 - 6√2

Okay, guys, let's bring it all together! We've simplified 6√√2 to 6⁴√2 and 4√2 - 10√2 to -6√2. Now our expression looks like this: 6⁴√2 - 6√2.

At this point, you might be tempted to combine these terms further, but hold on! Remember the rule about combining like terms: they need to have the same radical and the same index. In this case, we have ⁴√2 and √2. The indices are different (4 and 2, respectively), so we can't combine these terms directly. It's like trying to add apples and oranges – they're both fruits, but you can't say you have a single category of “fruit” when they're distinct types.

So, what does this mean for our final answer? Well, it means that 6⁴√2 - 6√2 is the simplest form of the expression. We've done all the simplification we can do. There are no more like terms to combine, and we've expressed each term in its simplest radical form.

It's important to recognize when you've reached the simplest form. Sometimes, the urge to simplify further can lead to incorrect steps. In this case, understanding the rules for combining radicals helps us see that we've reached the end of the road.

This final step highlights the importance of paying attention to detail and understanding the nuances of mathematical rules. We've taken a complex-looking expression and, through careful step-by-step simplification, arrived at a clear and concise answer. This process not only helps us solve the problem but also reinforces our understanding of radicals and algebraic manipulation.

So, there you have it! We've successfully simplified 6√√2 + 4√2 - 10√2 to 6⁴√2 - 6√2. We broke down the problem, tackled each part individually, and then brought it all together. Hopefully, this step-by-step guide has made the process clear and straightforward. Now you can confidently approach similar problems and simplify them like a pro! Remember, practice makes perfect, so keep working on those radical expressions, and you'll become a master in no time!

Final Simplified Expression

After walking through each step, we've arrived at our final simplified expression. The initial expression 6√√2 + 4√2 - 10√2 has been simplified to 6⁴√2 - 6√2. This is the most reduced form we can achieve given the terms and their respective radicals.

Conclusion

Simplifying mathematical expressions, especially those involving radicals, might seem challenging at first. However, by breaking the problem down into smaller, manageable steps, we can tackle even the most complex expressions. In this guide, we've seen how to simplify 6√√2 + 4√2 - 10√2 by first understanding the basics of radicals, then simplifying individual terms, and finally combining like terms where possible.

Remember, the key to success in mathematics is understanding the underlying principles and applying them systematically. Don't be afraid to take your time, break problems down, and practice regularly. With each problem you solve, you'll build confidence and improve your skills. So, keep practicing, and you'll become a math whiz in no time! And remember, we started with simplifying 6√√2 + 4√2 - 10√2 and nailed it!