Simplifying √50 + √72 + √18: A Detailed Solution
Hey guys! Today, we're going to dive deep into simplifying the expression √50 + √72 + √18. This might seem a bit daunting at first, but don't worry, we'll break it down step by step so it's super easy to understand. Math can be fun, especially when we tackle these problems together! So, let's get started and unlock the secrets of simplifying radicals.
Understanding the Basics of Simplifying Radicals
Before we jump into the main problem, let's quickly recap what it means to simplify a radical. Simplifying radicals essentially means expressing them in their simplest form. This often involves finding perfect square factors within the radicand (the number inside the square root symbol) and taking their square roots. Remember, the goal is to make the number inside the square root as small as possible. This not only makes the expression easier to work with but also helps in comparing and combining different radical expressions.
Why do we simplify radicals? You might wonder why we even bother simplifying these expressions. Well, simplifying radicals is crucial for several reasons. First, it makes calculations easier. Imagine trying to add √50 and √72 directly versus adding their simplified forms. The latter is much simpler, right? Second, simplified radicals allow for easier comparison. You can quickly see relationships between numbers when they are in their most basic form. Finally, it's a standard practice in mathematics. Most instructors and textbooks expect you to present your answers with simplified radicals. So, let's get good at it!
Breaking Down the Numbers
To kick things off, we need to understand how to break down numbers into their factors, specifically looking for those perfect square factors we talked about. A perfect square is a number that is the result of squaring an integer (e.g., 4, 9, 16, 25, etc.). Identifying these factors is key to simplifying radicals. For instance, if you have √20, you can break 20 down into 4 * 5, where 4 is a perfect square. This allows you to simplify √20 as 2√5. Mastering this skill is like having a superpower in the world of radicals!
The Golden Rule of Square Roots
Here’s a golden rule to remember: √(a * b) = √a * √b. This rule is the foundation of simplifying radicals. It tells us that the square root of a product is the product of the square roots. This allows us to separate perfect square factors from the rest of the radicand. For example, √50 can be written as √(25 * 2), which can then be separated into √25 * √2. Since √25 is 5, we get 5√2. This is the magic behind simplifying radicals, and it’s a trick you’ll use over and over again.
Step-by-Step Solution for √50 + √72 + √18
Alright, with the basics covered, let's tackle our main problem: √50 + √72 + √18. We're going to take it one step at a time, making sure each step is crystal clear. Remember, the key is to break down each radical into its simplest form and then combine any like terms. This is where the fun begins!
Step 1: Simplify √50
Let's start with √50. We need to find the largest perfect square that divides 50. Think about it – what perfect square goes into 50? If you guessed 25, you're spot on! 50 can be written as 25 * 2. Now we can rewrite √50 as √(25 * 2). Using our golden rule, we separate this into √25 * √2. And what's √25? It's 5! So, √50 simplifies to 5√2. See? Not so scary, right?
Step 2: Simplify √72
Next up, we have √72. This one might seem a bit trickier, but let's break it down. What’s the largest perfect square that divides 72? You might think of 9, which is a good start, but there's an even larger one: 36! 72 can be written as 36 * 2. So, √72 becomes √(36 * 2). Separating this gives us √36 * √2. And √36 is 6, so √72 simplifies to 6√2. We're on a roll!
Step 3: Simplify √18
Now, let's simplify √18. What’s the largest perfect square that divides 18? It's 9! We can write 18 as 9 * 2. So, √18 becomes √(9 * 2), which separates into √9 * √2. Since √9 is 3, √18 simplifies to 3√2. We've conquered another one!
Step 4: Combine Like Terms
Here comes the final step: combining like terms. We've simplified each radical, and now we have 5√2 + 6√2 + 3√2. Notice something? They all have the same radical part, √2. This means they are like terms, and we can add their coefficients (the numbers in front of the √2). So, we add 5 + 6 + 3, which equals 14. Therefore, 5√2 + 6√2 + 3√2 equals 14√2. And that’s our final simplified answer!
Common Mistakes to Avoid
Before we wrap up, let's talk about some common mistakes people make when simplifying radicals. Avoiding these pitfalls will help you nail these problems every time. One frequent mistake is not finding the largest perfect square factor. For example, with √72, someone might break it down into √(9 * 8) instead of √(36 * 2). While you can still simplify √(9 * 8) further, it adds extra steps. Always aim for the largest perfect square to save time and effort. Another mistake is incorrectly adding radicals. Remember, you can only add like radicals – those with the same radicand. You can't add √2 and √3 directly, for instance. Stick to these guidelines, and you'll be simplifying radicals like a pro!
Practice Problems
Now that we've walked through the solution, let’s solidify your understanding with a few practice problems. Try simplifying these on your own:
- √27 + √48
- √80 - √20
- 2√12 + 3√75
Work through these step by step, and you'll build confidence in your skills. Remember to look for the largest perfect square factors and combine like terms. Practice makes perfect, and the more you practice, the easier it will become.
Conclusion
So, there you have it! We’ve successfully simplified √50 + √72 + √18 and covered the essential techniques for simplifying radicals. Remember, the key is to break down each radical into its simplest form by finding perfect square factors and then combining like terms. Simplifying radicals is a fundamental skill in math, and mastering it will help you tackle more complex problems with ease. Keep practicing, stay curious, and you'll become a math whiz in no time! Keep up the great work, guys! You've got this!
