Simplifying Radicals: Product Of Surds Explained

Have you ever stumbled upon an expression involving radicals and wondered how to simplify it? This article dives deep into the process of simplifying the product of surds, specifically focusing on the expression $(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$. We'll break down each step, making it easy to understand and apply to similar problems. Let's explore how to tackle such mathematical challenges!

Understanding the Problem: Multiplying Surds

When you're faced with a mathematical problem like this, the first thing to do is understand what it's asking. In this case, we need to find the product of two expressions, each involving square roots, also known as surds. The expression is $(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$. To solve this, we'll use the distributive property, a fundamental concept in algebra. This property, often remembered by the acronym FOIL (First, Outer, Inner, Last), helps us multiply two binomials (expressions with two terms). Before we dive into the detailed steps, it’s crucial to remember some fundamental properties of radicals. Specifically, recall that $\sqrt{a} * \sqrt{b} = \sqrt{ab}$, which is crucial for simplifying the product of square roots. Understanding this rule will greatly assist in breaking down the complexities of the given expression. Recognizing this foundational principle allows us to simplify the terms and combine like radicals effectively. Let's take a closer look at how we apply these concepts in simplifying our target expression. The process involves not only applying the distributive property but also identifying opportunities to simplify individual radicals, making the final result as concise as possible. Furthermore, maintaining accuracy in every step is essential to prevent errors and achieve the correct solution. With a methodical approach and attention to detail, simplifying this type of expression becomes a manageable and rewarding mathematical exercise.

Step-by-Step Solution: Applying the Distributive Property

To solve the product $(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$, we'll apply the distributive property. This means each term in the first parenthesis is multiplied by each term in the second parenthesis. Let's break it down:

1. First: Multiply the first terms: $\sqrt{14} * \sqrt{12}$.
2. Outer: Multiply the outer terms: $\sqrt{14} * \sqrt{7}$.
3. Inner: Multiply the inner terms: $-\sqrt{3} * \sqrt{12}$.
4. Last: Multiply the last terms: $-\sqrt{3} * \sqrt{7}$.

Now, let’s execute each step. First, we multiply the first terms: $\sqrt{14} * \sqrt{12}$. Using the property $\sqrt{a} * \sqrt{b} = \sqrt{ab}$, this becomes $\sqrt{14 * 12} = \sqrt{168}$. We’ll simplify this later. Next, we multiply the outer terms: $\sqrt{14} * \sqrt{7} = \sqrt{14 * 7} = \sqrt{98}$. Moving on to the inner terms, we have $-\sqrt{3} * \sqrt{12} = -\sqrt{3 * 12} = -\sqrt{36}$. Finally, multiplying the last terms gives us $-\sqrt{3} * \sqrt{7} = -\sqrt{3 * 7} = -\sqrt{21}$. Now, let’s assemble these results. We have $\sqrt{168} + \sqrt{98} - \sqrt{36} - \sqrt{21}$. The next crucial step involves simplifying each radical individually. This involves finding perfect square factors within each radicand (the number under the square root). Simplifying the radicals will lead us to combining like terms and obtaining the final simplified expression. This step-by-step approach ensures that we don’t miss any terms and that we correctly apply the distributive property, setting the stage for the subsequent simplification process.

Simplifying Radicals: Finding Perfect Square Factors

In this crucial step, we simplify each radical term obtained in the previous section: $\sqrt{168}$, $\sqrt{98}$, $-\sqrt{36}$, and $-\sqrt{21}$. Simplifying radicals involves identifying and extracting perfect square factors from within the square root. This process makes the expression more concise and easier to understand.

Let’s begin with $\sqrt{168}$. We look for perfect square factors of 168. We can write 168 as $4 * 42$. Thus, $\sqrt{168} = \sqrt{4 * 42} = \sqrt{4} * \sqrt{42} = 2\sqrt{42}$. Next, we simplify $\sqrt{98}$. We can express 98 as $49 * 2$. Therefore, $\sqrt{98} = \sqrt{49 * 2} = \sqrt{49} * \sqrt{2} = 7\sqrt{2}$. Now, consider $-\sqrt{36}$. Since 36 is a perfect square ($6^2$), we have $-\sqrt{36} = -6$. Lastly, we look at $-\sqrt{21}$. The number 21 does not have any perfect square factors other than 1, so $-\sqrt{21}$ remains as it is. Now that we have simplified each radical, we can rewrite our expression: $2\sqrt{42} + 7\sqrt{2} - 6 - \sqrt{21}$. We have successfully broken down each radical into its simplest form, paving the way for the final combination of like terms. Identifying perfect square factors is a key skill in simplifying radicals, allowing us to express square roots in a more manageable and concise manner. This methodical approach ensures that no radical is left unsimplified, and we are one step closer to the solution.

Combining Like Terms: The Final Simplification

Now that we've simplified each radical, we have the expression $2\sqrt{42} + 7\sqrt{2} - 6 - \sqrt{21}$. The next step is to combine any like terms. Like terms in this context are terms that have the same radical part. For instance, $2\sqrt{42}$ and $5\sqrt{42}$ would be like terms because they both have $\sqrt{42}$. However, $2\sqrt{42}$ and $7\sqrt{2}$ are not like terms because they have different radicals.

Looking at our expression, we have terms with $\sqrt{42}$, $\sqrt{2}$, a constant term, and a term with $\sqrt{21}$. Let's examine each term: $2\sqrt{42}$ has no like terms in the expression. $7\sqrt{2}$ also has no like terms. The constant term -6 is a standalone term. Lastly, $-\sqrt{21}$ has no like terms either. Since there are no like terms to combine, our expression is already in its simplest form. Thus, the simplified form of the product $(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$ is $2\sqrt{42} + 7\sqrt{2} - 6 - \sqrt{21}$. This meticulous approach to combining like terms ensures that we do not inadvertently add or subtract terms that shouldn’t be combined, leading to an accurate final simplified expression. By recognizing and grouping like terms effectively, we can present the solution in its most concise and understandable form, showcasing a clear understanding of radical simplification.

The Answer: Putting It All Together

After carefully applying the distributive property, simplifying radicals, and attempting to combine like terms, we've arrived at the simplified expression: $2\sqrt{42} + 7\sqrt{2} - 6 - \sqrt{21}$. Now, we compare this result with the given options to find the correct answer.

Looking back at the options:

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. $2 \sqrt{42}-\sqrt{21}$

We can clearly see that our simplified expression matches option A. Therefore, the correct answer is $2 \sqrt{42}+7 \sqrt{2}-6-\sqrt{21}$. This methodical comparison with the provided options ensures that we confidently select the answer that aligns perfectly with our derived solution. By systematically working through the simplification process and double-checking our result against the choices, we demonstrate a thorough understanding of the concepts and techniques involved in simplifying radical expressions. This careful approach minimizes the risk of error and reinforces the accuracy of our final answer, showcasing a comprehensive mastery of the problem.

Conclusion

Simplifying expressions involving radicals might seem daunting at first, but by breaking the problem down into manageable steps, it becomes much more approachable. We started by applying the distributive property (FOIL), then simplified each radical by finding perfect square factors, and finally, we combined like terms. In this case, the product $(\sqrt{14}-\sqrt{3})(\sqrt{12}+\sqrt{7})$ simplifies to $2 \sqrt{42}+7 \sqrt{2}-6-\sqrt{21}$. Remember, practice is key to mastering these skills. The more you work with radicals, the more comfortable you'll become with simplifying them. Keep practicing, and you'll find these problems becoming second nature. For further learning and practice, you can explore resources like Khan Academy's Algebra section on radicals.