Simplifying Radicals: A Step-by-Step Guide

Welcome to the world of radicals! Today, we're diving deep into the simplification of expressions involving square roots. Our focus will be on solving the expression: $-5 imes \sqrt{63} + 2 \times \sqrt{7} + \sqrt{28}$. This might seem intimidating at first, but with a clear understanding of the rules and a bit of practice, you'll be simplifying radical expressions like a pro. This guide will break down the process step-by-step, making it easy to follow along. We will cover the key concepts, provide detailed explanations, and walk through the calculations to ensure you grasp every aspect. Let's embark on this mathematical journey together! To begin, let's clarify what a radical is and how it relates to our expression. A radical, represented by the symbol $\sqrt{}$, indicates the square root of a number. Simplifying a radical expression involves rewriting it in its simplest form, where the number under the radical sign (the radicand) has no perfect square factors other than 1. This means we aim to extract any perfect squares from within the radicals, leaving the remaining factors inside the radical. Our expression combines several radicals, each with a different radicand. The main goal here is to simplify each radical individually before combining the terms. This approach ensures that we are working with the most reduced forms of the radicals, making the final simplification process cleaner and more efficient. Remember that the ability to simplify radicals is a fundamental skill in algebra and is essential for solving various types of equations and problems. Let’s get started and break this down step-by-step.

Understanding the Basics: Radicals and Perfect Squares

Before we jump into the expression, let's get our foundations right. Understanding radicals and perfect squares is key to simplifying radical expressions. A radical is simply another term for a root, particularly a square root in this context. The expression under the radical sign is called the radicand. The goal of simplification is to remove any perfect square factors from the radicand. A perfect square is a number that results from squaring an integer. For example, 9 is a perfect square because $3^2 = 9$. Similarly, 16 is a perfect square because $4^2 = 16$. Knowing your perfect squares is super helpful in simplifying radicals. The perfect squares you should know include 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on. Now, let’s revisit our expression: $-5 \times \sqrt{63} + 2 \times \sqrt{7} + \sqrt{28}$. Our objective is to simplify each of the radical terms individually. To do this, we need to identify the perfect square factors within the radicands. This step involves factoring the numbers under the square root and identifying the largest perfect square factor. The process allows us to separate the radicals into the product of a perfect square and another number, allowing us to simplify them efficiently. The process of identifying perfect squares is crucial, as it’s the gateway to simplifying. Let's practice by identifying the perfect square factors in our expression. In $\sqrt{63}$, the perfect square factor is 9 ($9 \times 7 = 63$). For $\sqrt{28}$, the perfect square factor is 4 ($4 \times 7 = 28$). Notice that $\sqrt{7}$ does not have any perfect square factors other than 1, so it is already in its simplest form. Understanding these fundamental concepts will enable you to navigate through the simplification process with confidence.

Step-by-Step Simplification: Breaking Down the Expression

Now, let's get to the main course: the step-by-step simplification of our radical expression. We'll start by breaking down each radical term. First, let's address $-5 \times \sqrt{63}$. As we identified before, 63 can be factored into $9 \times 7$. Since 9 is a perfect square, we can rewrite the expression as $-5 \times \sqrt{9 \times 7}$. Using the property $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can further break this down to $-5 \times \sqrt{9} \times \sqrt{7}$. Since the square root of 9 is 3, this simplifies to $-5 \times 3 \times \sqrt{7}$, which equals $-15 \sqrt{7}$. This is the simplified form of our first term. Next, let's look at the second term, $2 \times \sqrt{7}$. This term is already in its simplest form because 7 has no perfect square factors other than 1. So, we'll just keep it as $2 \sqrt{7}$. Moving on to the third term, $\sqrt{28}$, we can factor 28 into $4 \times 7$. This allows us to rewrite $\sqrt{28}$ as $\sqrt{4 \times 7}$. Again, using the property $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can break it down to $\sqrt{4} \times \sqrt{7}$. Since the square root of 4 is 2, this simplifies to $2 \sqrt{7}$. Now, we have simplified each of the radical terms in the original expression. The next step is to combine these simplified terms. Let's summarise our progress. We started with $-5 \times \sqrt{63} + 2 \times \sqrt{7} + \sqrt{28}$. We then simplified each term individually to get $-15 \sqrt{7} + 2 \sqrt{7} + 2 \sqrt{7}$. Now, it's time to bring it all together.

Combining Like Terms: The Final Simplification

In the final step, we combine the simplified terms to arrive at the final simplified form of the expression. Remember, to combine radical terms, they must have the same radicand. In our simplified expression, all the terms have $\sqrt{7}$ as the radicand. The expression we're working with is $-15 \sqrt{7} + 2 \sqrt{7} + 2 \sqrt{7}$. To combine these, we simply add the coefficients (the numbers in front of the radical) while keeping the radical part the same. So, we have $-15 + 2 + 2$, which equals -11. Therefore, the simplified form of the expression is $-11 \sqrt{7}$. This is our final answer. Congratulations, you've successfully simplified the radical expression! The process involves several key steps: identifying perfect square factors, rewriting the radicals using these factors, simplifying the square roots, and then combining like terms. This approach ensures that you systematically work through the expression, making it manageable and easy to solve. Throughout this guide, we've broken down each step to provide you with a clear understanding of the concepts involved. The more you practice, the more comfortable you'll become with simplifying radical expressions. Keep practicing, and you'll find that these problems become more straightforward with each attempt. Remember the key takeaway: Always aim to extract the perfect square factors from the radicands. Combining like terms is straightforward once each radical has been simplified. With this knowledge, you are well-equipped to tackle similar problems.

Conclusion: Mastering Radical Simplification

In this guide, we've successfully navigated the process of simplifying the expression $-5 \times \sqrt{63} + 2 \times \sqrt{7} + \sqrt{28}$, and the answer is $-11 \sqrt{7}$. We started by understanding the basics of radicals and perfect squares. We then moved on to the step-by-step simplification of the expression. We broke down each radical term, identified perfect square factors, simplified the square roots, and finally, combined like terms. The process may seem daunting at first, but with a solid grasp of the foundational concepts and consistent practice, you'll gain confidence and proficiency in simplifying radical expressions. Remember that simplifying radicals is an essential skill in mathematics and is frequently used in solving various equations and mathematical problems. Consistent practice and a thorough understanding of the principles of radicals and perfect squares will enable you to solve such problems easily. Each step in the process, from factoring to combining like terms, is critical. With each expression you tackle, you'll refine your understanding and improve your problem-solving skills. Don’t hesitate to practice more examples and seek out additional resources. The goal is not just to find the correct answer, but also to develop a deeper understanding of the underlying mathematical principles. Keep practicing, and you will become proficient in simplifying radical expressions.

For further learning, consider visiting the following websites:

• Khan Academy - Simplifying Radicals