Simplifying The Expression: $3rown{45} + 4rown{80} - 2rown{125} - rown{245}$

Hey guys! Today, we're diving into the world of simplifying radical expressions. Specifically, we're tackling the expression 3rown{45} + 4rown{80} - 2rown{125} - rown{245}. This might look a bit intimidating at first, but don't worry, we'll break it down step by step so it's super easy to understand. Our main goal here is to express each radical in its simplest form by factoring out perfect squares. This will allow us to combine like terms and get to our final simplified answer. So, grab your pencils and let's get started!

Breaking Down the Radicals

Alright, let's start by simplifying each radical term individually. This involves finding the largest perfect square that divides each number under the square root. Remember, a perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). This step is crucial because it allows us to rewrite the radicals in a simpler form. When we simplify radicals, we make the numbers inside the square root as small as possible, which makes calculations easier and the expression look cleaner.

Simplifying $\sqrt{45}$

Okay, so let's look at $\sqrt{45}$ first. We need to find the largest perfect square that divides 45. Can you think of any? Well, 45 can be written as 9 times 5, and 9 is a perfect square (3 squared). So, we can rewrite $\sqrt{45}$ as $\sqrt{9 \times 5}$. Now, remember the rule that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$? We're going to use that here. So, $\sqrt{9 \times 5}$ becomes $\sqrt{9} \times \sqrt{5}$. And since $\sqrt{9}$ is just 3, we have $3\sqrt{5}$. See? We've simplified $\sqrt{45}$ to $3\sqrt{5}$. This is much easier to work with!

Simplifying $\sqrt{80}$

Next up, let's tackle $\sqrt{80}$. What's the largest perfect square that divides 80? Well, 80 can be written as 16 times 5, and 16 is a perfect square (4 squared). So, we can rewrite $\sqrt{80}$ as $\sqrt{16 \times 5}$. Just like before, we can separate this into $\sqrt{16} \times \sqrt{5}$. And since $\sqrt{16}$ is 4, we get $4\sqrt{5}$. Awesome! $\sqrt{80}$ is now simplified to $4\sqrt{5}$.

Simplifying $\sqrt{125}$

Now, let's simplify $\sqrt{125}$. The largest perfect square that divides 125 is 25 (5 squared), since 125 is 25 times 5. So, we rewrite $\sqrt{125}$ as $\sqrt{25 \times 5}$. Separating this gives us $\sqrt{25} \times \sqrt{5}$, and since $\sqrt{25}$ is 5, we have $5\sqrt{5}$. Great job! $\sqrt{125}$ simplifies to $5\sqrt{5}$.

Simplifying $\sqrt{245}$

Lastly, let's simplify $\sqrt{245}$. This one might be a bit trickier, but we can do it! The largest perfect square that divides 245 is 49 (7 squared), because 245 is 49 times 5. So, we rewrite $\sqrt{245}$ as $\sqrt{49 \times 5}$. This becomes $\sqrt{49} \times \sqrt{5}$, and since $\sqrt{49}$ is 7, we end up with $7\sqrt{5}$. Perfect! $\sqrt{245}$ is simplified to $7\sqrt{5}$.

Substituting the Simplified Radicals

Now that we've simplified each radical individually, let's substitute them back into the original expression. This is where all our hard work starts to pay off! By replacing the original radicals with their simplified forms, we'll be able to combine like terms more easily. Remember, like terms are terms that have the same radical part. For example, $3\sqrt{5}$ and $4\sqrt{5}$ are like terms because they both have $\sqrt{5}$. Substituting the simplified radicals is a key step in getting to the final simplified expression.

So, our original expression was: $3\sqrt{45} + 4\sqrt{80} - 2\sqrt{125} - \sqrt{245}$.

We found that:

• $\sqrt{45} = 3\sqrt{5}$
• $\sqrt{80} = 4\sqrt{5}$
• $\sqrt{125} = 5\sqrt{5}$
• $\sqrt{245} = 7\sqrt{5}$

Now, let's plug these back into the original expression:

$3(3\sqrt{5}) + 4(4\sqrt{5}) - 2(5\sqrt{5}) - 7\sqrt{5}$

See how much simpler it looks already? We've replaced the larger radicals with their simplified versions, and now we're ready for the next step: simplifying the coefficients.

Simplifying the Coefficients

Before we can combine the terms, we need to simplify the coefficients (the numbers in front of the radicals). This involves performing the multiplication that we set up in the previous step. Remember, we're just multiplying the numbers outside the square roots. This is a straightforward process, but it's important to get it right so we don't make any mistakes. Once we've simplified the coefficients, we'll be ready to identify and combine the like terms. This is where the expression will really start to come together and we'll get closer to our final answer.

Let's go through each term one by one:

• $3(3\sqrt{5}) = 9\sqrt{5}$
• $4(4\sqrt{5}) = 16\sqrt{5}$
• $2(5\sqrt{5}) = 10\sqrt{5}$

So, our expression now looks like this:

$9\sqrt{5} + 16\sqrt{5} - 10\sqrt{5} - 7\sqrt{5}$

Much better, right? All the terms have the same radical part ($\sqrt{5}$), which means we can now combine them. Let's move on to the next step and do just that!

Combining Like Terms

Alright, we're in the home stretch now! We've simplified the radicals and the coefficients, and the only thing left to do is combine the like terms. Remember, like terms are terms that have the same radical part. In our case, all the terms have $\sqrt{5}$, so they're all like terms. To combine them, we simply add or subtract the coefficients. This is just like combining variables in algebra – we're just treating $\sqrt{5}$ as if it were a variable, like 'x'. Once we've combined the like terms, we'll have our final simplified expression. How cool is that?

Let's take a look at our expression again:

$9\sqrt{5} + 16\sqrt{5} - 10\sqrt{5} - 7\sqrt{5}$

Now, let's combine the coefficients:

$9 + 16 - 10 - 7 = ?$

Let's do the addition and subtraction step by step:

• $9 + 16 = 25$
• $25 - 10 = 15$
• $15 - 7 = 8$

So, the combined coefficient is 8. This means our simplified expression is:

$8\sqrt{5}$

Final Answer

And there you have it, guys! The simplified form of $3\sqrt{45} + 4\sqrt{80} - 2\sqrt{125} - \sqrt{245}$ is $8\sqrt{5}$. We did it! We took a complex-looking expression and broke it down into manageable steps. First, we simplified each radical individually by finding perfect square factors. Then, we substituted the simplified radicals back into the original expression. Next, we simplified the coefficients by performing the multiplication. Finally, we combined the like terms to arrive at our final answer. See, simplifying radicals isn't so scary after all! With a little practice, you'll be able to tackle these types of problems with ease.

Remember, the key is to take it one step at a time and stay organized. Write down each step clearly, and don't be afraid to double-check your work. And most importantly, have fun with it! Math can be like a puzzle, and it's so satisfying when you finally solve it. So, keep practicing and keep exploring the world of radicals. You've got this!