Multiplying Radicals: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of multiplying radicals. Specifically, we'll tackle an expression that might seem a little intimidating at first glance: ${\sqrt{6}(7 \sqrt{10}+4)}$. Don't worry, though; we'll break it down into easy-to-understand steps, ensuring you grasp the concept and can confidently solve similar problems in the future. So, grab your pencils, and let's get started. Multiplying radicals involves a few key principles, primarily the distributive property and the rules of simplifying radicals. The distributive property tells us that when we have an expression like a(b + c), we need to multiply 'a' by both 'b' and 'c'. In our case, ${\sqrt{6}}$ plays the role of 'a', ${7 \sqrt{10}}$ is 'b', and 4 is 'c'. Simplifying radicals, on the other hand, means rewriting a radical expression in its simplest form, where there are no perfect square factors inside the radical. This usually involves finding the largest perfect square that divides the number under the radical and then extracting the square root of that perfect square. Let's start with the basics, we'll go through the distributive property first, then focus on simplifying the radicals. This is one of those math concepts that, once you get the hang of it, feels super satisfying! Let's get into it.

Distributing the Radical: First Steps

Multiplying radicals and applying the distributive property is the first crucial step in simplifying the expression ${\sqrt{6}(7 \sqrt{10}+4)}$. This involves multiplying ${\sqrt{6}}$ by both terms inside the parentheses: ${7 \sqrt{10}}$ and 4. Remember, the distributive property is all about making sure every term inside the parentheses gets multiplied by the term outside. So, we'll perform the following multiplications: ${\sqrt{6} \times 7 \sqrt{10}}$ and ${\sqrt{6} \times 4}$. Let's break this down further. When multiplying ${\sqrt{6} \times 7 \sqrt{10}}$, we can rearrange the terms because of the commutative property of multiplication, which states that the order of multiplication doesn't change the result. We can rewrite this as ${7 \times \sqrt{6} \times \sqrt{10}}$. Next, multiply the constants (numbers without radicals): ${7}$. And then multiply the radicals. To multiply radicals, multiply the numbers under the radical signs: ${\sqrt{6} \times \sqrt{10} = \sqrt{6 \times 10} = \sqrt{60}}$. So, the first part of our multiplication becomes ${7\sqrt{60}}$. Now, let's look at the second part, ${\sqrt{6} \times 4}$. This is straightforward. Multiply the radical by the constant to get ${4\sqrt{6}}$. So, after distributing ${\sqrt{6}}$, our expression becomes ${7\sqrt{60} + 4\sqrt{6}}$. This initial distribution sets the stage for simplifying the radicals. Notice how we've expanded the original expression, which is a key step in simplifying the whole thing. Now it’s time to simplify the radicals. Keep going; we're doing great, guys!

Simplifying the Radicals: Making Things Simpler

Now, let's talk about simplifying the radicals. We've got ${7\sqrt{60} + 4\sqrt{6}}$, and our goal is to simplify both ${\sqrt{60}}$ and ${\sqrt{6}}$ as much as possible. Simplifying a radical involves finding the largest perfect square factor of the number under the radical and then extracting its square root. Let’s start with ${\sqrt{60}}$. We need to find the largest perfect square that divides 60. Perfect squares are numbers like 1, 4, 9, 16, 25, 36, and so on. Looking at the factors of 60 (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60), we see that 4 is a perfect square factor of 60. So, we can rewrite ${\sqrt{60}}$ as ${\sqrt{4 \times 15}}$. Now, because ${\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}}$, we can further simplify ${\sqrt{4 \times 15}}$ to ${\sqrt{4} \times \sqrt{15}}$. The square root of 4 is 2, so this becomes ${2\sqrt{15}}$. But we're not done yet, since the ${7\sqrt{60}}$ is multiplied by 7. We had ${7\sqrt{60}}$ and we now have ${7 \times 2\sqrt{15}}$, which is ${14\sqrt{15}}$. Now let's simplify ${\sqrt{6}}$. The factors of 6 are 1, 2, 3, and 6. There is no perfect square other than 1 that divides 6. Therefore, ${\sqrt{6}}$ is already in its simplest form. So our expression becomes ${14\sqrt{15} + 4\sqrt{6}}$. This is our final, simplified answer. Simplifying radicals makes the whole expression more manageable and easier to work with, both in this example and in more complex math problems. Keep up the good work! We're almost there, and you're doing great. High five!

Final Answer and Conclusion: Putting it All Together

Alright, guys! Let's look at the final answer and wrap things up. After distributing ${\sqrt{6}}$ and simplifying the radicals, we found that ${\sqrt{6}(7 \sqrt{10}+4)}$ simplifies to ${14\sqrt{15} + 4\sqrt{6}}$. This is our final answer, and it represents the most simplified form of the original expression. Remember, we started with ${\sqrt{6}(7 \sqrt{10}+4)}$. We first used the distributive property to get ${7\sqrt{60} + 4\sqrt{6}}$. Then, we simplified the radical ${\sqrt{60}}$ to ${2\sqrt{15}}$, making the term ${7\sqrt{60}}$ into ${14\sqrt{15}}$. Since ${\sqrt{6}}$ could not be simplified further, our final expression became ${14\sqrt{15} + 4\sqrt{6}}$. This process highlights the importance of understanding both the distributive property and the rules for simplifying radicals. Multiplying radicals might seem tricky at first, but with practice, you'll become more comfortable with the steps involved. Always remember to distribute, simplify, and ensure your answer is in its simplest form. Mastering these skills will not only help you in algebra but also lay a strong foundation for more advanced math concepts. Keep practicing, and you'll be multiplying radicals like a pro in no time! So, that's it for today's lesson. I hope you found it helpful and enjoyable. Keep practicing, and don't be afraid to ask questions. Math is all about practice and persistence. You've got this!