Simplifying Radicals What Is The Product Of (√14-√3)(√12+√7)

Hey there, math enthusiasts! Today, we're diving into a fascinating problem involving radicals. We're going to explore how to simplify the product $(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$. This type of problem often appears in algebra and pre-calculus, and mastering it will significantly boost your problem-solving skills. Let's break it down step by step, so you can confidently tackle similar challenges in the future. Get ready to sharpen your pencils and dive into the world of square roots!

Decoding the Expression: A Step-by-Step Simplification

Let's start our mathematical journey by diving headfirst into the heart of the problem. Our mission is to simplify the expression $(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$. At first glance, it might seem a bit daunting, but fear not! We'll conquer this beast using the distributive property, a fundamental tool in algebra. Think of it like this: we're going to methodically multiply each term in the first set of parentheses by each term in the second set. It's like a carefully choreographed dance of numbers and symbols, where each step brings us closer to the solution.

So, let's break it down: First, we'll multiply $\sqrt{14}$ by both terms in the second parenthesis, $\sqrt{12}$ and $\sqrt{7}$. Then, we'll do the same with $-\sqrt{3}$, multiplying it by $\sqrt{12}$ and $\sqrt{7}$. This process will give us a series of terms that we can then simplify and combine. Remember, the key here is to be meticulous and patient. Each step is crucial, and a steady hand will guide us to the correct answer. As we move through the multiplication process, we'll keep an eye out for opportunities to simplify the radicals. This might involve breaking down the numbers under the square roots into their prime factors and looking for pairs that can be brought out of the radical sign. This is where our knowledge of number theory and radical simplification comes into play.

Now, let's put this plan into action. Grab your pencils, and let's get started on this exciting mathematical adventure!

$\sqrt{14} * \sqrt{12}$ this is the first term that needs simplification. Remember the golden rule of radicals: $\sqrt{a} * \sqrt{b} = \sqrt{a*b}$. Applying this rule, we get $\sqrt{14*12}$ which simplifies to $\sqrt{168}$. But we're not done yet! We need to see if we can simplify the radical further by factoring 168. The prime factorization of 168 is $2^3 * 3 * 7$. We can rewrite $\sqrt{168}$ as $\sqrt{2^2 * 2 * 3 * 7}$, and since $\sqrt{2^2}$ is 2, we can pull that out of the radical, leaving us with $2\sqrt{2 * 3 * 7}$ which simplifies to $2\sqrt{42}$. Great job! We've simplified the first term.

Next, let's tackle $\sqrt{14} * \sqrt{7}$. Applying the same rule, we get $\sqrt{14*7}$ which is $\sqrt{98}$. Let's break down 98 into its prime factors: $98 = 2 * 7^2$. So, $\sqrt{98}$ can be rewritten as $\sqrt{2 * 7^2}$. We can pull the $7^2$ out of the radical as 7, leaving us with $7\sqrt{2}$. Fantastic! We're making excellent progress.

Now, let's move on to the third term: $-\sqrt{3} * \sqrt{12}$. This gives us $-\sqrt{3*12}$ which is $-\sqrt{36}$. And here's a delightful surprise: 36 is a perfect square! The square root of 36 is 6, so this term simplifies to -6. How satisfying is that?

Finally, we have $-\sqrt{3} * \sqrt{7}$, which gives us $-\sqrt{3*7}$, or $-\sqrt{21}$. This term is already in its simplest form, so we can move on.

Now, let's gather all our simplified terms: $2\sqrt{42} + 7\sqrt{2} - 6 - \sqrt{21}$. And there you have it! We've successfully simplified the original expression. But our journey isn't over yet. We need to see if we can combine any like terms. Remember, we can only combine terms that have the same radical part. In this case, we have $2\sqrt{42}$, $7\sqrt{2}$, -6, and $-\sqrt{21}$. None of these terms have the same radical, so we can't simplify further. This is our final answer. What a triumphant feeling!

Assembling the Simplified Expression

Alright guys, after carefully multiplying and simplifying each term, we've arrived at a pivotal moment. It's time to gather all the pieces of our mathematical puzzle and assemble the fully simplified expression. Remember, we started with $(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$ and meticulously worked our way through each multiplication. We transformed $\sqrt{14} * \sqrt{12}$ into the simplified form of $2\sqrt{42}$. Then, we expertly handled $\sqrt{14} * \sqrt{7}$, which became $7\sqrt{2}$. Next in line was $-\sqrt{3} * \sqrt{12}$, which gracefully simplified to -6. And finally, we tackled $-\sqrt{3} * \sqrt{7}$, leaving us with $-\sqrt{21}$.

Now, let's bring these simplified terms together like a harmonious ensemble of mathematical notes. We have $2\sqrt{42}$, $7\sqrt{2}$, -6, and $-\sqrt{21}$. The next crucial step is to combine these terms into a single expression. This is where we must exercise our keen eye for detail and ensure that each term is placed correctly. It's like arranging the instruments in an orchestra, where each one has its designated spot to create the most melodious sound. So, with precision and care, we combine our terms: $2\sqrt{42} + 7\sqrt{2} - 6 - \sqrt{21}$.

This expression represents the simplified form of our original product. It's the culmination of our hard work, our careful calculations, and our unwavering commitment to solving the problem. But before we declare victory, there's one final step we must take. We need to examine our expression to see if there are any like terms that can be combined. This is like the final polish on a masterpiece, ensuring that every detail is perfect.

Scrutinizing for Like Terms: The Final Polish

Okay, team, we're in the home stretch! We've assembled our simplified expression: $2\sqrt{42} + 7\sqrt{2} - 6 - \sqrt{21}$. Now comes the crucial step of scrutinizing for like terms. This is where we put on our detective hats and carefully examine each term to see if any of them share a common radical. It's like searching for matching puzzle pieces, where the radicals are the unique shapes that must fit together.

Remember, the golden rule for combining terms with radicals is that they must have the same radical part. This means that the number under the square root sign must be identical. For example, $3\sqrt{5}$ and $7\sqrt{5}$ are like terms because they both have $\sqrt{5}$. We can combine them by simply adding their coefficients: $3\sqrt{5} + 7\sqrt{5} = 10\sqrt{5}$. However, $3\sqrt{5}$ and $7\sqrt{2}$ are not like terms because they have different radicals, $\sqrt{5}$ and $\sqrt{2}$, respectively. We cannot combine these terms.

So, let's apply this rule to our expression. We have four terms: $2\sqrt{42}$, $7\sqrt{2}$, -6, and $-\sqrt{21}$. The first term, $2\sqrt{42}$, has a radical of $\sqrt{42}$. The second term, $7\sqrt{2}$, has a radical of $\sqrt{2}$. The third term, -6, is a constant and doesn't have a radical. And the fourth term, $-\sqrt{21}$, has a radical of $\sqrt{21}$.

As we carefully compare the radicals, we notice something important: none of the terms have the same radical part. $\sqrt{42}$, $\sqrt{2}$, and $\sqrt{21}$ are all distinct radicals. This means that we cannot combine any of these terms. It's like having puzzle pieces that simply don't fit together. And the constant term, -6, is also in a league of its own, as it doesn't have any radical part to match with.

Therefore, after our diligent search, we conclude that there are no like terms in our expression. This means that our expression is already in its simplest form. We've reached the end of our simplification journey! Give yourselves a pat on the back, mathletes, because you've conquered this challenge with flying colors.

The Final Answer: A Moment of Triumph

Drumroll, please! After our meticulous journey through multiplication, simplification, and the careful search for like terms, we've finally arrived at the final answer. It's a moment of triumph, where all our hard work culminates in a single, elegant expression. We started with the product $(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$ and, through the power of the distributive property and our mastery of radicals, we've transformed it into its simplest form.

Our final expression, the result of our mathematical prowess, is: $2\sqrt{42} + 7\sqrt{2} - 6 - \sqrt{21}$. This is the simplified form of the original product, and it represents the culmination of our efforts. It's like reaching the summit of a challenging mountain, where the view is breathtaking and the sense of accomplishment is overwhelming.

But what does this answer truly mean? It means that we've successfully navigated the complexities of radical expressions and emerged victorious. We've demonstrated our understanding of the distributive property, our ability to simplify radicals, and our keen eye for identifying like terms. These are valuable skills that will serve us well in future mathematical endeavors. More than just an answer, this expression is a testament to our problem-solving abilities and our dedication to the pursuit of mathematical knowledge.

And now, the moment you've all been waiting for: Let's compare our simplified expression to the answer choices provided. We have:

A. $2 \sqrt{42}+7 \sqrt{2}-6-\sqrt{21}$ B. $\sqrt{14}-6+\sqrt{7}$ C. $\sqrt{26}+\sqrt{21}-\sqrt{15}-\sqrt{10}$ D. $2 \sqrt{42}-\sqrt{21}$

With a confident smile, we can see that our simplified expression, $2\sqrt{42} + 7\sqrt{2} - 6 - \sqrt{21}$, perfectly matches answer choice A. We've done it! We've successfully simplified the product and identified the correct answer. This is a moment to celebrate our mathematical achievement. Give yourselves a hearty round of applause, because you've earned it!

Why Option A is the Ultimate Winner

Alright, let's break down why option A is the star of the show and the ultimate winner in our quest to simplify $(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$. We've already done the heavy lifting, meticulously working through the multiplication and simplification steps. But it's always a good idea to solidify our understanding by revisiting the key reasons why our answer, $2 \sqrt{42}+7 \sqrt{2}-6-\sqrt{21}$, reigns supreme.

The first and most crucial reason is that our answer directly results from the correct application of the distributive property and the rules of radical simplification. We carefully multiplied each term in the first parenthesis by each term in the second parenthesis, ensuring that no term was left behind. It was like conducting a thorough search, making sure we accounted for every mathematical element. Then, we expertly simplified each resulting term, breaking down radicals into their simplest forms. This involved factoring numbers under the square root and identifying perfect square factors that could be brought out of the radical sign. This meticulous process guaranteed that our simplified terms were accurate and in their most elegant form.

But the magic didn't stop there! We then gathered our simplified terms and combined them into a single expression. This is where our attention to detail truly shone. We made sure to maintain the correct signs and coefficients for each term. And finally, we performed the crucial step of checking for like terms. This is where we put on our critical thinking hats and carefully compared the radicals in each term. We knew that only terms with the same radical could be combined. And after our thorough examination, we confidently concluded that there were no like terms in our expression.

This entire process, from the initial multiplication to the final check for like terms, was executed with precision and accuracy. And the result is our triumphant expression: $2 \sqrt{42}+7 \sqrt{2}-6-\sqrt{21}$. This expression perfectly matches option A, solidifying its position as the correct answer.

Dissecting the Distractors: Why Other Options Fall Short

Now, let's put on our detective hats and delve into the world of incorrect answer choices. Understanding why the other options are wrong is just as important as knowing why the correct answer is right. It helps us solidify our understanding of the concepts and avoid common pitfalls in the future. So, let's dissect the distractors and uncover their mathematical missteps.

First, let's take a look at option B: $\sqrt{14}-6+\sqrt{7}$. At first glance, this option might seem tempting because it contains some of the numbers we encountered during our simplification process. However, a closer inspection reveals that it's missing some crucial terms and doesn't accurately represent the result of our multiplication and simplification. It's like a puzzle with missing pieces, leaving a fragmented and incomplete picture. The absence of terms like $2\sqrt{42}$ and $7\sqrt{2}$ immediately raises a red flag, indicating that this option is not the correct answer.

Next up is option C: $\sqrt{26}+\sqrt{21}-\sqrt{15}-\sqrt{10}$. This option appears to be a mishmash of various square roots, with no clear connection to our original expression or the steps we took to simplify it. It's like a random assortment of ingredients that don't quite come together to form a coherent dish. The numbers under the square roots (26, 21, 15, and 10) don't align with the numbers we encountered during our simplification process, further solidifying its incorrectness. This option seems to stem from a misunderstanding of how radicals combine during multiplication and simplification.

Finally, we have option D: $2 \sqrt{42}-\sqrt{21}$. This option is closer to the correct answer than options B and C, as it contains the terms $2\sqrt{42}$ and $-\sqrt{21}$, which are present in our simplified expression. However, it's still missing the crucial term $7\sqrt{2}$ and the constant term -6. It's like a dish with some of the key ingredients but lacking the essential elements that bring it all together. The absence of these terms indicates that this option likely results from an incomplete simplification process, where some terms were overlooked or incorrectly combined.

In conclusion, each of the incorrect options falls short for different reasons. Option B is missing crucial terms, option C is a jumble of unrelated square roots, and option D is an incomplete simplification. By understanding these shortcomings, we reinforce our understanding of the correct simplification process and develop a keen eye for spotting errors.

Wrapping Up: Mastering the Art of Simplifying Radicals

And that, my friends, brings us to the end of our exciting journey through the world of simplifying radical expressions! We've successfully tackled the problem $(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$, and in doing so, we've not only arrived at the correct answer but also honed our skills in a variety of crucial mathematical techniques. This is what it’s all about, guys – not just getting the right answer, but truly understanding the process.

We started with a seemingly complex expression, but we didn't let that intimidate us. Instead, we broke it down into manageable steps, applying the distributive property with precision and care. We multiplied each term, keeping a watchful eye on signs and coefficients. Then, we dove into the world of radical simplification, factoring numbers under the square root and identifying perfect square factors. This allowed us to extract those factors from the radical and express our terms in their simplest forms. And let's be honest, there's something incredibly satisfying about seeing a messy radical transform into an elegant, simplified term!

But our journey didn't stop there. We then gathered our simplified terms and faced the crucial task of checking for like terms. This is where our critical thinking skills truly came into play. We remembered the golden rule: only terms with the same radical part can be combined. And with a careful comparison of radicals, we confidently concluded that our expression was already in its simplest form.

Throughout this process, we also developed a valuable skill in analyzing incorrect answer choices. We dissected the distractors, uncovering the mathematical missteps that led to their incorrectness. This not only reinforced our understanding of the correct solution but also helped us develop a keen eye for spotting errors in the future.

So, what's the key takeaway from all of this? It's that mastering the art of simplifying radicals is not just about memorizing formulas and rules. It's about developing a deep understanding of the underlying concepts, a meticulous approach to problem-solving, and a willingness to break down complex problems into manageable steps. With these skills in your mathematical arsenal, you'll be well-equipped to tackle any radical expression that comes your way. Keep practicing, keep exploring, and keep pushing your mathematical boundaries. The world of numbers is vast and fascinating, and there's always something new to discover!

So, keep up the amazing work, guys! You're on the path to becoming true mathematical masters!