Solving 12/√3 - 8/√3 + 1 A Step-by-Step Mathematical Guide
Hey guys! Let's dive into solving this interesting mathematical problem: 12/√3 - 8/√3 + 1. It looks a bit tricky at first glance, but don't worry, we'll break it down step by step so it's super easy to understand. This article is designed to guide you through each stage of the solution, ensuring you grasp the underlying concepts and can tackle similar problems with confidence. We'll cover everything from simplifying radicals to performing basic arithmetic operations, making math feel less like a chore and more like a fun puzzle. So, grab your thinking caps, and let's get started!
Understanding the Basics
Before we jump into the solution, let's brush up on some fundamental mathematical concepts that will help us along the way. Understanding these basics is crucial, guys, because they form the building blocks for solving more complex problems. We'll touch on radicals, rationalizing denominators, and basic arithmetic operations. Think of it as preparing our toolkit before we start the actual work. Knowing these concepts inside and out will not only help you solve this particular problem but also equip you with skills applicable to a wide range of mathematical challenges. So, let's make sure we're all on the same page before we move forward!
Radicals and Square Roots
First off, what exactly is a radical? In simple terms, a radical is a symbol (√) that indicates the root of a number. The most common type of radical is the square root, which asks: "What number, when multiplied by itself, equals the number under the radical?" For example, √9 is 3 because 3 * 3 = 9. Radicals can sometimes seem intimidating, but they're really just a way of expressing roots. Understanding radicals is essential for simplifying expressions and solving equations, especially those involving fractions with radicals in the denominator, as we'll see in our problem. Getting comfortable with radicals opens up a whole new world of mathematical possibilities!
Rationalizing the Denominator
Now, let's talk about rationalizing the denominator. This might sound like a mouthful, but it's a pretty straightforward process. It involves eliminating radicals from the denominator of a fraction. Why do we do this? Well, it's generally considered good mathematical practice to present fractions in their simplest form, and having a radical in the denominator isn't considered simple. The trick to rationalizing the denominator is to multiply both the numerator and the denominator by a form of 1 that will eliminate the radical in the denominator. For example, if you have a fraction with √3 in the denominator, you would multiply both the numerator and the denominator by √3. This process ensures that you're not changing the value of the fraction, just its appearance. Mastering this technique is super handy for simplifying expressions and making them easier to work with.
Basic Arithmetic Operations
Last but not least, let's quickly review basic arithmetic operations: addition, subtraction, multiplication, and division. These are the bread and butter of mathematics, and we'll be using them extensively to solve our problem. Remember the order of operations (often remembered by the acronym PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Keeping these operations in mind will help us avoid errors and solve the problem efficiently. So, with our toolkit ready, let's dive into the solution!
Step-by-Step Solution
Okay, guys, now that we've got our basics covered, let's tackle the problem at hand: 12/√3 - 8/√3 + 1. We're going to take this step by step, making sure each part is crystal clear. Our goal is to simplify the expression as much as possible and arrive at a final answer. The key here is to approach the problem methodically, breaking it down into smaller, more manageable chunks. So, let's roll up our sleeves and get started!
Step 1: Rationalizing the Denominators
The first thing we need to do is rationalize the denominators in the fractions 12/√3 and 8/√3. Remember, this means getting rid of the square root in the denominator. To do this, we'll multiply both the numerator and the denominator of each fraction by √3. This won't change the value of the fraction, but it will change its appearance, making it easier to work with. So, for 12/√3, we multiply both the top and bottom by √3, and we do the same for 8/√3. This is a crucial step because it sets us up for simpler calculations later on. By removing the radicals from the denominators, we're making the expression much more user-friendly.
For 12/√3:
(12/√3) * (√3/√3) = (12√3) / 3
For 8/√3:
(8/√3) * (√3/√3) = (8√3) / 3
Step 2: Simplifying the Fractions
Now that we've rationalized the denominators, let's simplify the fractions we've created. We have (12√3) / 3 and (8√3) / 3. Notice that both fractions have a common denominator, which is great news for us! But before we combine them, let's see if we can simplify each fraction individually. Look at (12√3) / 3. Can we divide 12 by 3? Absolutely! 12 divided by 3 is 4, so (12√3) / 3 simplifies to 4√3. Similarly, for (8√3) / 3, we can't simplify it as easily because 8 is not perfectly divisible by 3, so we'll leave it as it is for now. Simplifying fractions is all about finding common factors and reducing the numbers to their smallest possible form. This step makes our expression even cleaner and easier to handle.
(12√3) / 3 = 4√3
(8√3) / 3 remains as (8√3) / 3
Step 3: Combining Like Terms
Alright, guys, we're making great progress! Now that we've simplified our fractions, let's combine the like terms. Our expression now looks like this: 4√3 - (8√3) / 3 + 1. We have two terms with √3 and a constant term (1). To combine the terms with √3, we need to subtract (8√3) / 3 from 4√3. To do this, we'll first express 4√3 as a fraction with a denominator of 3. Think of it like adding or subtracting fractions – you need a common denominator! Once we have a common denominator, we can easily combine the √3 terms. This step is crucial for bringing all the pieces together and getting closer to our final answer. So, let's find that common denominator and combine those terms!
To subtract (8√3) / 3 from 4√3, we need a common denominator. We can rewrite 4√3 as (12√3) / 3.
Now we have: (12√3) / 3 - (8√3) / 3 + 1
Step 4: Performing the Subtraction
Now comes the fun part – performing the subtraction! We have (12√3) / 3 - (8√3) / 3. Since they have the same denominator, we can simply subtract the numerators. So, 12√3 minus 8√3 gives us 4√3. This means our expression simplifies to (4√3) / 3. We're really whittling down the problem now, guys! Subtracting fractions with common denominators is a straightforward process, and it's a key skill in simplifying expressions. By performing this subtraction, we've reduced our expression to a much more manageable form.
(12√3) / 3 - (8√3) / 3 = (4√3) / 3
Step 5: Adding the Constant
We're almost there, guys! Our expression is now (4√3) / 3 + 1. The final step is to add the constant. We have a fraction with a radical and a whole number. To add them together, we can think of 1 as a fraction with a denominator of 3, just like we did before. So, 1 can be written as 3/3. Now we can add the two terms together. This final addition will give us our answer in the simplest form possible. Remember, adding a constant is like adding apples and oranges – we need to make sure we're adding them in a way that makes sense mathematically. By expressing 1 as a fraction with the same denominator, we can seamlessly combine it with the other term.
(4√3) / 3 + 1 = (4√3) / 3 + 3/3
Step 6: Final Result
And there you have it! Our final result is (4√3 + 3) / 3. We've taken the original expression, broken it down step by step, and simplified it as much as possible. This is the answer to our problem: 12/√3 - 8/√3 + 1 = (4√3 + 3) / 3. High five, guys! We did it! This final result is a testament to our methodical approach and our understanding of the underlying mathematical principles. We've not only solved the problem but also reinforced our skills in handling radicals, rationalizing denominators, and performing basic arithmetic operations. Pat yourselves on the back – you've earned it!
(4√3) / 3 + 3/3 = (4√3 + 3) / 3
Conclusion
So, guys, we've successfully navigated the mathematical waters and found our solution: (4√3 + 3) / 3. We started with a seemingly complex expression and, by breaking it down into manageable steps, arrived at a simplified answer. We revisited the importance of rationalizing denominators, simplifying fractions, and combining like terms. Remember, the key to solving any mathematical problem is to approach it methodically and understand the basic principles involved. Math might seem daunting at times, but with a little practice and a step-by-step approach, you can conquer any challenge. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!
