Solving 2√6 Multiplied By 3√18 A Step-by-Step Guide

Hey guys! Let's dive into a fun math problem today: multiplying radicals! We're going to break down how to solve 2√6 * 3√18 step by step. Don't worry, it's not as scary as it looks. We'll make sure it's super clear and easy to follow, so you can tackle similar problems with confidence. So, grab your pencils and let’s get started!

Breaking Down the Problem

Okay, so the problem we're tackling is 2√6 * 3√18. The key to multiplying radicals like these is to remember a couple of fundamental rules. First, we can multiply the numbers outside the square roots together, and then we can multiply the numbers inside the square roots together. It’s like handling the coefficients and the radicals separately before combining them back. This makes the whole process much more manageable.

Let’s start with the numbers outside the square roots. We have 2 and 3. Multiplying these together is straightforward: 2 * 3 = 6. Easy peasy! Now, let’s move on to the trickier part – the square roots. We have √6 and √18. Remember, when multiplying radicals, we can multiply the numbers inside the square roots. So, we need to multiply 6 and 18. What does that give us? Well, 6 * 18 = 108. So now we have √108. We're halfway there!

Putting it all together, we have 6√108. But we're not quite done yet. The next step is crucial: simplifying the radical. Often, the number inside the square root can be broken down into factors, where at least one of them is a perfect square. This will help us to simplify the radical and get our final answer in the simplest form. So, let’s dive into simplifying √108.

Simplifying the Radical √108

Now, let's get down to simplifying √108. This is where things get interesting! Simplifying radicals means finding the largest perfect square that divides evenly into the number under the square root. In our case, we need to find the largest perfect square that divides 108. Perfect squares are numbers like 4, 9, 16, 25, 36, and so on – numbers that are the result of squaring an integer. So, we're looking for the biggest of these that fits into 108.

One way to do this is to start listing the factors of 108. Let's see: 108 can be divided by 2, giving us 54. It can be divided by 3, giving us 36. Aha! 36 is a perfect square because 6 * 6 = 36. So, we can rewrite 108 as 36 * 3. This is a crucial step because it allows us to simplify the square root much more easily. Now we can rewrite √108 as √(36 * 3). Remember, the goal here is to break down the number under the square root into factors that include a perfect square, making simplification possible.

Using the property of radicals that √(a * b) = √a * √b, we can split √(36 * 3) into √36 * √3. This is where the magic happens! We know that √36 is simply 6 because 36 is a perfect square. So now we have 6 * √3. That’s a lot simpler, right? We’ve taken √108 and simplified it to 6√3. Now we can take this simplified radical and plug it back into our original problem.

So, we've taken a potentially intimidating radical and broken it down into manageable parts. By identifying perfect square factors, we've made it much easier to work with. This technique is super useful for all sorts of problems involving radicals, so make sure you’re comfortable with it. Next, we’ll put this simplified form back into our main equation and wrap things up!

Putting It All Together

Alright, guys, let's bring it all together! We started with the expression 2√6 * 3√18, and we've made some serious progress. We figured out that multiplying the numbers outside the square roots (2 and 3) gives us 6. Then, we multiplied the numbers inside the square roots (6 and 18) to get √108. We rewrote our expression as 6√108.

But we didn't stop there! We knew we could simplify √108. We broke it down into √(36 * 3) and simplified it to √36 * √3, which then became 6√3. That was a crucial step, and now we're ready to finish the job. So, we have 6 from the numbers outside the square roots, and we have 6√3 from simplifying √108. Now, we need to multiply these two together. This means we’re multiplying 6 by 6√3.

When multiplying a whole number by a term with a radical, we multiply the whole numbers together and leave the radical part as it is. So, 6 * 6√3 becomes (6 * 6)√3. And what is 6 * 6? It’s 36! So, our final expression is 36√3. This is the simplified form of our original problem, 2√6 * 3√18.

We've taken a somewhat complex problem and broken it down into manageable steps. We multiplied the coefficients, multiplied the radicals, simplified the resulting radical, and then put it all back together. And there you have it! The final answer is 36√3. Awesome work! This methodical approach is key to successfully tackling radical multiplication problems.

Final Answer: 36√3

So, there you have it! The solution to 2√6 * 3√18 is 36√3. We walked through each step, from multiplying the coefficients and radicals to simplifying the final expression. Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. By simplifying radicals and understanding how to multiply them, you can tackle even the trickiest problems with confidence.

We started by multiplying the numbers outside the radicals (2 and 3) and the numbers inside the radicals (6 and 18). This gave us 6√108. Then, we focused on simplifying √108. We found that 108 can be factored into 36 * 3, where 36 is a perfect square. This allowed us to rewrite √108 as √(36 * 3), which simplifies to 6√3. Finally, we multiplied the 6 we got from the coefficients by the 6√3 we got from simplifying the radical, giving us our final answer of 36√3.

I hope this explanation was clear and helpful! Remember to practice these steps with different problems, and you’ll become a pro at multiplying radicals in no time. Keep up the great work, and don't hesitate to tackle more math challenges. You've got this!