Simplify $\sqrt{75} + \sqrt{48} - \sqrt{27}$: A Step-by-Step Guide

Hey Math Whizzes! Let's Tackle Square Roots Together!

Alright, awesome math adventurers! Today, we're diving deep into the fascinating world of simplifying square root expressions, and trust me, it's gonna be a blast. We're talking about expressions that might look a bit intimidating at first glance, like our main challenge for today: $\sqrt{75} + \sqrt{48} - \sqrt{27}$. If you've ever stared at these kinds of problems and wondered, "How on earth do I even begin?" then you're in the absolute right place. Many students find radicals a bit tricky, but with a solid game plan and a friendly guide (that's me!), you'll be zipping through them like a pro. Our goal isn't just to find the answer to this specific problem, but to truly understand the underlying principles so you can conquer any similar challenge that comes your way. Think of it as building your mathematical superpowers! We're going to break down this complex-looking problem into super manageable, easy-to-follow steps. By the end of this article, you won't just know the answer; you'll understand the why and the how, giving you the confidence to tackle more advanced topics down the line. We’re not just simplifying numbers; we're unlocking their potential! This isn't just about memorizing a formula; it's about grasping a fundamental concept in algebra that pops up everywhere, from geometry to physics. So, grab your favorite snack, get comfy, and let's embark on this exciting mathematical journey together to simplify square root expressions like a boss. We'll explore each part of the problem, discuss the strategies needed, and make sure every single one of you feels super confident about simplifying radicals. It’s all about building that foundation, piece by piece, so you're not just guessing but truly understanding the logic behind every move. This skill is incredibly valuable, not only for acing your math tests but also for developing a sharper, more analytical mind. Let's get this done, guys!

The Secret Sauce: Understanding Simplified Square Root Form

Before we jump into our specific problem, let's talk about the absolute secret sauce to mastering square roots: understanding what simplified square root form actually means. Imagine you have a messy room; you want to tidy it up, right? Simplifying a square root is pretty much the same idea. We want to get the number under the square root symbol (that's called the radicand) as small as possible, with no perfect square factors left inside. What's a perfect square factor, you ask? Great question! A perfect square is simply the result of multiplying an integer by itself – like 4 (2x2), 9 (3x3), 16 (4x4), 25 (5x5), 36 (6x6), and so on. When we're simplifying radicals, our main mission is to find the largest perfect square that divides evenly into the radicand. If a square root has a perfect square factor hiding inside, it's not considered simplified. For instance, $\sqrt{12}$ isn't simplified because 4 (which is $2^2$) is a factor of 12. We can rewrite $\sqrt{12}$ as $\sqrt{4 \times 3}$, and since we know $\sqrt{4}$ is 2, it becomes $2\sqrt{3}$. See? Much cleaner! The number 3 under the radical in $2\sqrt{3}$ has no perfect square factors (other than 1, which doesn't help us simplify further), so it's in its simplest form. This process is crucial for simplifying square root expressions because it allows us to combine them later, just like combining 'x' terms in an algebraic equation. If you have $2\sqrt{3}$ and $5\sqrt{3}$, you can easily add them to get $7\sqrt{3}$. But you can't easily add $\sqrt{12}$ and $\sqrt{75}$ until they are both in their simplified form with the same radical. Mastering this step is like learning the alphabet before you write a novel; it's fundamental. We'll be applying this concept rigorously to each term in our problem: $\sqrt{75}$, $\sqrt{48}$, and $\sqrt{27}$. So, get ready to unleash your inner detective and spot those hidden perfect square factors! This foundational understanding is truly what separates those who just get by from those who genuinely excel at math. It’s a powerful tool, guys!

Step 1: Breaking Down Each Square Root into Its Simplest Form

Alright, team, now that we understand the goal of simplified square root form, let's put that knowledge into action! We're going to tackle each term in our expression $\sqrt{75} + \sqrt{48} - \sqrt{27}$ one by one, giving them the full simplification treatment. This is where the magic happens, and each number gets its makeover. Remember, our mantra is to find the largest perfect square factor inside each radicand. Don't worry if it takes a moment to spot them; practice makes perfect, and soon you'll be seeing them everywhere!

Simplifying $\sqrt{75}$

Let's kick things off with simplifying square root of 75. We need to think: what's the biggest perfect square that divides evenly into 75? Let's list some perfect squares: 4, 9, 16, 25, 36, 49, 64... Can 75 be divided by 4? No. By 9? No. By 16? No. Ah, but wait! 75 divided by 25 gives us 3! And 25 is a perfect square ($5 \times 5$). Bingo! That's exactly what we're looking for. So, we can rewrite $\sqrt{75}$ like this:

• First, we express 75 as a product of its largest perfect square factor and another number: $75 = 25 \times 3$.
• Next, we substitute this back into our square root: $\sqrt{75} = \sqrt{25 \times 3}$.
• Now, thanks to a handy property of square roots, we can split this into two separate square roots: $\sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}$.
• And finally, we know that $\sqrt{25}$ is simply 5. So, $\sqrt{75}$ simplifies to $5\sqrt{3}$.

See how neat that is? We've extracted the perfect square, leaving a much smaller, un-simplifiable number under the radical. This step is super important for our overall goal of simplifying square root expressions!

Simplifying $\sqrt{48}$

Moving on to our next term, let's focus on simplifying square root of 48. Again, we're on the hunt for the largest perfect square factor of 48. Let's check our perfect squares: 4, 9, 16, 25, 36... Does 48 divide by 4? Yes, $48 \div 4 = 12$. So we could write $\sqrt{4 \times 12} = 2\sqrt{12}$. But wait! Is 12 simplified? No, it still has a perfect square factor (4) inside it! This is why it's crucial to find the largest perfect square. Let's try 16. Does 48 divide by 16? Yes! $48 \div 16 = 3$. And 16 is a perfect square ($4 \times 4$). Awesome! This is the one we want. Here's how we simplify it:

• We write 48 as the product of 16 and 3: $48 = 16 \times 3$.
• Substitute this into the square root: $\sqrt{48} = \sqrt{16 \times 3}$.
• Split it up: $\sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3}$.
• Calculate $\sqrt{16}$, which is 4. So, $\sqrt{48}$ simplifies to $4\sqrt{3}$.

Another one bites the dust! We're making excellent progress on our mission to simplify square root expressions. Notice a pattern emerging? Both $\sqrt{75}$ and $\sqrt{48}$ simplified to something times $\sqrt{3}$. This is a huge clue for what's coming next!

Simplifying $\sqrt{27}$

Last but not least for this step, we've got simplifying square root of 27. Time to find that largest perfect square factor for 27. Let's run through our list again: 4, 9, 16, 25... Does 27 divide by 4? No. Does it divide by 9? You bet it does! $27 \div 9 = 3$. And 9 is a perfect square ($3 \times 3$). Perfect! We've found our factor.

• Express 27 as $9 \times 3$.
• Plug it into the square root: $\sqrt{27} = \sqrt{9 \times 3}$.
• Separate the roots: $\sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3}$.
• Since $\sqrt{9}$ is 3, $\sqrt{27}$ simplifies to $3\sqrt{3}$.

Look at that! All three terms, $\sqrt{75}$, $\sqrt{48}$, and $\sqrt{27}$, have now been transformed into their simplest forms: $5\sqrt{3}$, $4\sqrt{3}$, and $3\sqrt{3}$, respectively. This is a critical milestone in our journey to simplify square root expressions. If you got these three steps down, you're practically a master already! This careful, methodical approach ensures accuracy and builds a strong understanding, which is what we're all about here. Don't rush this stage; it's the foundation of everything that follows, and getting it right here makes the rest of the problem a breeze. So take a moment, pat yourself on the back, and let's get ready for the final combination!

Step 2: Combining Like Terms – The Final Showdown!

Alright, my math enthusiasts, we've done the heavy lifting of simplifying each individual square root! Now comes the super satisfying part: combining square root terms. This is where all our hard work pays off and the solution truly takes shape. Remember our original problem: $\sqrt{75} + \sqrt{48} - \sqrt{27}$. After all that awesome simplification in Step 1, we transformed it into something much more manageable. Let's write down what we found:

• $\sqrt{75}$ became $5\sqrt{3}$
• $\sqrt{48}$ became $4\sqrt{3}$
• $\sqrt{27}$ became $3\sqrt{3}$

So, our original expression $\sqrt{75} + \sqrt{48} - \sqrt{27}$ can now be rewritten as $5\sqrt{3} + 4\sqrt{3} - 3\sqrt{3}$. Isn't that much friendlier looking? The key concept here is understanding like radicals. Just like you can add $5x + 4x - 3x$ because they all have the same variable 'x', you can add and subtract square roots only if they have the exact same number under the radical symbol. In our case, all our simplified terms share that wonderful $\sqrt{3}$! This makes them like radicals, and that means we can combine their coefficients (the numbers in front of the square root) just like any other numerical operation. Think of $\sqrt{3}$ as a common factor, almost like it's a special unit. We have 5 units of $\sqrt{3}$, then we add 4 more units of $\sqrt{3}$, and finally, we take away 3 units of $\sqrt{3}$. It’s super straightforward when you look at it that way. Let’s do the math with the coefficients:

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

Now, let's just perform the simple addition and subtraction inside the parentheses:

$5 + 4 = 9$

$9 - 3 = 6$

So, when we combine everything, we get $6\sqrt{3}$. And there you have it, folks! The final, fully simplified answer to $\sqrt{75} + \sqrt{48} - \sqrt{27}$ is $6\sqrt{3}$. This part of adding and subtracting square roots is often where people make mistakes if they haven't simplified properly beforehand, or if they try to combine unlike radicals. But because we were so meticulous in Step 1, this final step becomes a piece of cake. This whole process of simplifying square root expressions not only gives us the correct answer but also helps us appreciate the elegance and consistency of mathematical rules. It's truly empowering to take a complex-looking problem and break it down into simple, manageable steps, leading you right to the correct solution with confidence. You've earned this moment of mathematical triumph!

Comparing Our Answer with the Options

Fantastic work, everyone! We've meticulously walked through the process of simplifying square root expressions and arrived at our final answer: $6\sqrt{3}$. Now, let's take a look back at the original problem's options and see which one matches our hard-earned solution. This is the moment of truth, where we get to confirm our calculations and feel that sweet satisfaction of a job well done. The initial question provided these options:

• a) $8\sqrt{3}$
• b) $\sqrt{3}$
• c) $6\sqrt{3}$
• d) $\sqrt{96}$

As you can clearly see, our calculated answer, $6\sqrt{3}$, perfectly aligns with option c) $6\sqrt{3}$. Boom! Mission accomplished! It's always a great feeling when your detailed work leads you directly to one of the provided choices. But let's take a quick moment to understand why the other options aren't correct, especially for those curious minds out there. Option a) $8\sqrt{3}$ would imply a different sum of coefficients (if we had $5\sqrt{3} + 4\sqrt{3} - 1\sqrt{3}$ for example, or if one of the initial simplifications was off). Option b) $\sqrt{3}$ is too small; it would mean the coefficients summed up to 1, which clearly didn't happen in our calculation $(5+4-3=6)$. Now, let's consider option d) $\sqrt{96}$. This one is interesting because it's still in an unsimplified form. To truly compare it, we should try checking answers for square root simplification by simplifying $\sqrt{96}$ as well. What's the largest perfect square factor of 96? Let's see: $96 \div 16 = 6$. So, $\sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6}$. As you can see, $4\sqrt{6}$ is definitely not $6\sqrt{3}$. The numbers under the radical are different, meaning they are unlike radicals and cannot be considered equivalent to our answer. This reinforces the importance of expressing all radicals in their simplest form to avoid confusion and ensure accurate comparisons. So, the process of simplifying square root problem options can sometimes involve checking each choice to confirm. Our method was sound, and the match is undeniable. Great job identifying the correct answer and understanding why the others fall short!

Why Master Square Root Simplification? (Beyond the Classroom!)

"Okay, I get it," you might be thinking, "I can simplify $\sqrt{75} + \sqrt{48} - \sqrt{27}$. But why does this even matter outside of a math textbook?" That's an excellent question, my friends! And the answer is, simplifying radicals is far from just an academic exercise; it's a fundamental skill with surprising real-world applications across various fields. Understanding the importance of simplifying radicals extends well beyond getting a good grade on your next math test. Think about it: whenever we deal with measurements that aren't perfectly neat, or when we apply geometric principles, square roots pop up constantly. For example, in geometry, if you're using the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the length of a hypotenuse or the diagonal of a square, your answer might initially come out as a square root like $\sqrt{72}$ or $\sqrt{125}$. To communicate these lengths clearly and precisely, especially if you need to combine them with other measurements or perform further calculations, you must simplify them. $\sqrt{72}$ is $6\sqrt{2}$, and $6\sqrt{2}$ is much easier to work with than a big, clunky $\sqrt{72}$. In physics and engineering, quantities like electrical impedance, wave frequencies, and even certain relativistic effects often involve square roots. Engineers designing structures or calculating forces might encounter radical expressions that need to be simplified to ensure accuracy and efficiency in their formulas and computations. Imagine an architect needing to calculate exact distances or areas; they won't want to work with $\sqrt{180}$ when they can use $6\sqrt{5}$. It makes calculations cleaner and reduces the chance of errors. Even in computer graphics and game development, understanding how to manipulate numbers efficiently, including radicals, can be crucial for optimizing algorithms that render complex shapes and calculate distances in 3D spaces. Real-world square root applications also extend to fields like finance (calculating standard deviation or volatility), statistics, and even music theory (relating frequencies and intervals). Essentially, simplifying radicals allows us to present mathematical information in its most elegant, understandable, and useful form. It's about clarity, precision, and making complex calculations more manageable. So, by mastering this skill, you're not just solving a math problem; you're equipping yourself with a powerful tool for analytical thinking and problem-solving that will serve you well in countless situations, both inside and outside the classroom. Keep practicing, because these skills are truly building blocks for future success!

Wrapping It Up: Your Journey to Square Root Mastery

Wow, guys, what a fantastic journey we've had! We started with an expression that looked a bit daunting, $\sqrt{75} + \sqrt{48} - \sqrt{27}$, and through a clear, step-by-step process, we've totally conquered it. You've now seen exactly how to simplify square root expressions from start to finish. Let's do a quick recap of the most important takeaways from our adventure to square root mastery:

1. Identify Perfect Square Factors: The first and most crucial step is always to break down each individual square root by finding the largest perfect square factor within its radicand. This is the core of simplifying. Remember our list: 4, 9, 16, 25, 36, etc.
2. Extract and Simplify: Once you've found that perfect square, pull it out from under the radical by taking its square root. So, $\sqrt{25 \times 3}$ becomes $5\sqrt{3}$. This leaves the smallest possible whole number under the radical.
3. Combine Like Radicals: After all your terms are simplified, you can only add or subtract them if they have the exact same number under the square root symbol (these are called like radicals). You then just add or subtract their coefficients, leaving the common radical as is.

For those of you who want to get even better at how to get better at radicals, here are some pro tips: Practice makes perfect! The more you work with perfect squares and prime factorization, the quicker you'll be at spotting those hidden factors. Try to memorize the first few perfect squares (up to 100 or even 144) – it'll speed up your calculations immensely. And always, always double-check your work! A small error in simplification can throw off your entire answer. This skill is a true asset in your mathematical toolkit, opening doors to understanding more complex algebra and geometry. You've shown that with a methodical approach and a little bit of patience, even the most intimidating math problems can be broken down and solved. Keep that mathematical curiosity alive, and remember that every problem you solve makes you a stronger, more confident learner. You rock!