Multiplying Radicals: A Step-by-Step Guide

Are you struggling with multiplying radical expressions? Fear not! This comprehensive guide will walk you through the process step-by-step, using the example $2 \sqrt{3}(\sqrt{15}+\sqrt{7})$ as our main focus. By the end of this article, you'll have a solid understanding of how to tackle similar problems with confidence. So, let’s dive into the world of radicals and multiplication!

Understanding Radicals

Before we jump into the multiplication process, let's make sure we have a clear understanding of what radicals are. A radical is a mathematical expression that uses a root, such as a square root, cube root, or higher. The most common radical is the square root, denoted by the symbol ${\sqrt{}}$. Understanding radicals is crucial, as they represent numbers that, when multiplied by themselves, give you the number under the root. For example, ${\sqrt{9} = 3}$ because 3 * 3 = 9. Similarly, in our problem, we have ${\sqrt{3}}$, ${\sqrt{15}}$, and ${\sqrt{7}}$, each representing a number that, when squared, results in 3, 15, and 7, respectively. In the expression $2 \sqrt{3}(\sqrt{15}+\sqrt{7})$, the numbers 3, 15, and 7 under the square root symbol are called radicands. The number 2 outside the first radical is a coefficient, indicating that the radical ${\sqrt{3}}$ is being multiplied by 2. When dealing with radical expressions, it's essential to follow the order of operations, which dictates that we handle multiplication and distribution before moving on to any addition or subtraction within the expression. This foundational understanding will help you tackle more complex problems involving radicals with ease. Remember, the key is to break down the problem into manageable steps, and that's exactly what we'll do as we proceed with multiplying these radicals.

The Distributive Property: Our Key Tool

At the heart of multiplying radical expressions lies the distributive property. This fundamental property of mathematics states that a(b + c) = ab + ac. In simpler terms, it means you can multiply a single term by each term inside a set of parentheses and then add the results. When multiplying radicals, the distributive property allows us to break down complex expressions into simpler ones. Consider our problem, $2 \sqrt{3}(\sqrt{15}+\sqrt{7})$. Here, $2 \sqrt{3}$ is the term outside the parentheses, and $(\sqrt{15}+\sqrt{7})$ is the expression inside the parentheses. Applying the distributive property, we need to multiply $2 \sqrt{3}$ by both ${\sqrt{15}}$ and ${\sqrt{7}}$ separately. This transforms our single, more complex expression into two simpler multiplication problems: $(2 \sqrt{3} * \sqrt{15}) + (2 \sqrt{3} * \sqrt{7})$. By using this approach, we've effectively divided the original problem into smaller, more manageable parts. This step is crucial because it sets the stage for simplifying the radicals individually before combining them, if possible. The distributive property is a cornerstone of algebra, and mastering its application in scenarios like this is vital for success in simplifying and solving various types of mathematical problems. Now that we've used the distributive property to expand our expression, let's move on to the next step: multiplying the radicals themselves.

Multiplying the Radicals: Step-by-Step

Now that we've distributed, let's focus on multiplying the individual radical terms. The key rule here is that you can multiply the coefficients (the numbers outside the radical) together and the radicands (the numbers inside the radical) together. Let's break down the first part of our expression: $2 \sqrt{3} * \sqrt{15}$. Here, we have a coefficient of 2 for the first term and an implied coefficient of 1 for the second term (since ${\sqrt{15}}$ is the same as $1 * \sqrt{15}$). Multiplying the coefficients gives us 2 * 1 = 2. Next, we multiply the radicands: 3 * 15 = 45. So, $2 \sqrt{3} * \sqrt{15}$ becomes $2 \sqrt{45}$. Now, let’s tackle the second part of our expression: $2 \sqrt{3} * \sqrt{7}$. Again, we multiply the coefficients: 2 * 1 = 2. Then, we multiply the radicands: 3 * 7 = 21. This gives us $2 \sqrt{21}$. So, our expression now looks like this: $2 \sqrt{45} + 2 \sqrt{21}$. Remember, when multiplying radicals, the product of two radicals is the radical of the product. By systematically multiplying coefficients and radicands, we transform the initial complex expression into a more manageable form. This step is crucial because it prepares us for the next phase: simplifying the radicals. Simplification not only makes the expression cleaner but also helps us identify any like terms that can be combined. Let's proceed to the simplification process to further refine our answer.

Simplifying Radicals: Finding Perfect Squares

The next crucial step in solving our problem is simplifying the radicals. Simplifying radicals involves finding perfect square factors within the radicand (the number under the square root) and extracting their square roots. Let's revisit our expression: $2 \sqrt{45} + 2 \sqrt{21}$. First, consider $2 \sqrt{45}$. We need to find a perfect square that divides 45. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25). In this case, 45 can be factored as 9 * 5, where 9 is a perfect square (3 * 3 = 9). We can rewrite $2 \sqrt{45}$ as $2 \sqrt{9 * 5}$. Using the property ${\sqrt{a * b} = \sqrt{a} * \sqrt{b}}$, we can further break this down into $2 * \sqrt{9} * \sqrt{5}$. Since ${\sqrt{9} = 3}$, we get $2 * 3 * \sqrt{5}$, which simplifies to $6 \sqrt{5}$. Now, let's look at the second term, $2 \sqrt{21}$. We need to determine if 21 has any perfect square factors. The factors of 21 are 1, 3, 7, and 21. None of these (other than 1) are perfect squares, so ${\sqrt{21}}$ cannot be simplified further. Therefore, $2 \sqrt{21}$ remains as it is. Simplifying radicals is a critical skill in dealing with radical expressions. It allows us to express the radicals in their simplest form, making them easier to work with and compare. Now that we have simplified each term individually, our expression is $6 \sqrt{5} + 2 \sqrt{21}$. The final step is to check if we can combine any like terms, but before we do that, let's understand what like terms are in the context of radicals.

Combining Like Terms: Are They Present?

The final step in simplifying our expression is to check for like terms. In the context of radicals, like terms are terms that have the same radicand (the number under the square root). Let's take a look at our expression: $6 \sqrt{5} + 2 \sqrt{21}$. The first term has ${\sqrt{5}}$, and the second term has ${\sqrt{21}}$. Since the radicands (5 and 21) are different, these terms are not like terms. Consequently, we cannot combine them. If we had, for instance, $6 \sqrt{5} + 3 \sqrt{5}$, we could combine these because they both have ${\sqrt{5}}$. We would simply add the coefficients (the numbers in front of the radical), resulting in $(6 + 3) \sqrt{5} = 9 \sqrt{5}$. However, in our current expression, $6 \sqrt{5} + 2 \sqrt{21}$, the radicals are different, so no further simplification is possible through combining like terms. Recognizing and combining like terms is a crucial skill in simplifying radical expressions, as it helps to reduce the expression to its most compact form. In this specific case, because our terms are not alike, our simplified expression remains as is. This underscores the importance of carefully examining the radicands before attempting to combine terms. With that, we’ve reached the final form of our expression.

The Final Answer

After carefully following all the steps, we've arrived at our final answer. We started with the expression $2 \sqrt{3}(\sqrt{15}+\sqrt{7})$, applied the distributive property, multiplied the radicals, simplified each term, and checked for like terms. Our journey through this problem has demonstrated the importance of understanding and applying each of these steps methodically. Let's recap: We first distributed $2 \sqrt{3}$ across the terms inside the parentheses, resulting in $2 \sqrt{3} * \sqrt{15} + 2 \sqrt{3} * \sqrt{7}$. Then, we multiplied the coefficients and radicands to get $2 \sqrt{45} + 2 \sqrt{21}$. Next, we simplified ${\sqrt{45}}$ to $3 \sqrt{5}$, giving us $2 * 3 \sqrt{5}$, which further simplified to $6 \sqrt{5}$. The term $2 \sqrt{21}$ could not be simplified further because 21 has no perfect square factors. Finally, we checked for like terms but found none, as ${\sqrt{5}}$ and ${\sqrt{21}}$ are different radicals. Therefore, our final, simplified answer is: $6 \sqrt{5} + 2 \sqrt{21}$. This result represents the most reduced form of the original expression, and it showcases the power of breaking down complex problems into smaller, manageable steps. Remember, mastering these steps will allow you to confidently tackle similar radical multiplication problems in the future. For further learning on simplifying radical expressions, you might find helpful resources on websites like Khan Academy.