Multiplying & Simplifying: (2+√3)(1+4√6) Guide

Have you ever stumbled upon an expression like $(2+\sqrt{3})(1+4 \sqrt{6})$ and felt a little intimidated? Don't worry; you're not alone! Multiplying and simplifying radical expressions might seem tricky at first, but with the right approach, it becomes a breeze. This article will break down the process step-by-step, ensuring you not only understand how to solve this specific problem but also gain the skills to tackle similar challenges. Let’s dive in and demystify the world of radical multiplication and simplification. Remember, mastering these techniques opens doors to more advanced mathematical concepts, making your journey through mathematics smoother and more rewarding. Understanding the core principles is key, so let's start building that foundation together.

Understanding the Basics of Radical Expressions

Before we jump into the multiplication process, let's quickly recap what radical expressions are and how they behave. A radical expression is simply an expression that contains a square root, cube root, or any other root. In our case, we're dealing with square roots, denoted by the symbol $\sqrt{ }$. Understanding the properties of square roots is crucial for simplifying expressions. For instance, the product of square roots can be combined under a single radical: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. Similarly, we need to remember how to handle multiplication involving both rational numbers and radicals. This foundation will help us methodically approach the multiplication and simplification of $(2+\sqrt{3})(1+4 \sqrt{6})$. Mastering these basics is like having the right tools for a job; it makes the entire process much easier and more efficient. So, before we move on, make sure you're comfortable with these fundamental concepts – they're the building blocks of everything we'll be doing next. Let's ensure we're all on the same page before we proceed further.

The Distributive Property: Our Key Tool

The distributive property is a fundamental concept in algebra that allows us to multiply a sum by multiplying each addend separately and then adding the products. In simpler terms, it states that a(b + c) = ab + ac. This property is our main tool for expanding expressions like $(2+\sqrt{3})(1+4 \sqrt{6})$. We'll apply the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last) to ensure that each term in the first parenthesis is multiplied by each term in the second parenthesis. This systematic approach guarantees we don't miss any multiplications. The distributive property isn't just a mathematical rule; it's a method of breaking down complex problems into smaller, manageable steps. By using this property, we transform a seemingly complicated expression into a series of simpler multiplications and additions. So, let's keep this powerful tool in mind as we proceed, because it's the key to unlocking the simplification of our radical expression.

Step-by-Step Multiplication of (2+√3)(1+4√6)

Now, let's put the distributive property into action and multiply the expression $(2+\sqrt{3})(1+4 \sqrt{6})$. We'll follow the FOIL method, ensuring we multiply each term correctly:

1. First: Multiply the first terms in each parenthesis: 2 * 1 = 2
2. Outer: Multiply the outer terms: 2 * $4\sqrt{6} = 8\sqrt{6}$
3. Inner: Multiply the inner terms: $\sqrt{3}$ * 1 = $\sqrt{3}$
4. Last: Multiply the last terms: $\sqrt{3}$ * $4\sqrt{6} = 4\sqrt{18}$

Combining these results, we get: $2 + 8\sqrt{6} + \sqrt{3} + 4\sqrt{18}$. This is an expanded form of our original expression, but we're not done yet. The next crucial step is simplification, where we'll look for ways to reduce the radicals and combine like terms. Remember, each step in this process builds upon the previous one. So, by carefully applying the distributive property, we've successfully expanded the expression and set the stage for the next phase: simplification. Let's move forward with confidence and continue our journey towards the final simplified answer.

Simplifying the Radicals

The expression we've obtained, $2 + 8\sqrt{6} + \sqrt{3} + 4\sqrt{18}$, contains a radical that can be further simplified: $4\sqrt{18}$. To simplify this, we need to find the largest perfect square that divides 18. We recognize that 18 can be written as 9 * 2, and 9 is a perfect square (3 * 3). Therefore, we can rewrite $\sqrt{18}$ as $\sqrt{9 * 2}$. Using the property of square roots, we can separate this into $\sqrt{9} * \sqrt{2}$, which simplifies to $3\sqrt{2}$. Now, we substitute this back into our expression: $4\sqrt{18} = 4 * 3\sqrt{2} = 12\sqrt{2}$. Our expression now looks like this: $2 + 8\sqrt{6} + \sqrt{3} + 12\sqrt{2}$. Simplifying radicals is a critical step because it ensures that our answer is in its most reduced form. By identifying and extracting perfect square factors, we make the expression cleaner and easier to work with. This skill is not just useful for this particular problem; it's a fundamental technique in algebra that will serve you well in various mathematical contexts. So, let's take a moment to appreciate the power of simplification and move on to the final step: combining like terms.

Combining Like Terms

Now that we've simplified the radicals, our expression is $2 + 8\sqrt{6} + \sqrt{3} + 12\sqrt{2}$. The next step is to identify and combine any like terms. In this case, we have one constant term (2) and three radical terms: $8\sqrt{6}$, $\sqrt{3}$, and $12\sqrt{2}$. Notice that none of these radical terms have the same radicand (the number under the square root symbol). This means they are not like terms and cannot be combined. Therefore, the expression $2 + 8\sqrt{6} + \sqrt{3} + 12\sqrt{2}$ is already in its simplest form. It's important to remember that we can only combine terms that have the same radical part. Just like we can't add apples and oranges, we can't directly add terms with different radicands. This step highlights the importance of careful observation and understanding the rules of combining terms. By recognizing that there are no like terms to combine, we can confidently conclude that we've reached the final simplified answer. Combining like terms is the final polish on our work, ensuring that our answer is as concise and clear as possible.

The Final Simplified Answer

After carefully multiplying, simplifying, and combining like terms, we arrive at our final answer: $2 + 8\sqrt{6} + \sqrt{3} + 12\sqrt{2}$. This expression cannot be simplified further, as there are no more like terms to combine and all radicals are in their simplest form. This journey through multiplying and simplifying radicals demonstrates the power of step-by-step problem-solving. By breaking down a complex expression into manageable parts, we were able to systematically apply mathematical principles and arrive at a clear, concise answer. This process not only provides the solution to this particular problem but also equips us with the skills and confidence to tackle similar challenges in the future. Remember, mathematics is often about the journey of discovery as much as it is about the final destination. So, let's celebrate our accomplishment and carry this newfound knowledge forward.

In conclusion, mastering the multiplication and simplification of radical expressions is a valuable skill in mathematics. By understanding the distributive property, simplifying radicals, and combining like terms, we can confidently tackle these types of problems. The final simplified form of $(2+\sqrt{3})(1+4 \sqrt{6})$ is $2 + 8\sqrt{6} + \sqrt{3} + 12\sqrt{2}$.

For further exploration and practice on radical expressions, consider visiting Khan Academy's Algebra section.