Simplify $(\sqrt{12}+\sqrt{6})(\sqrt{6}-\sqrt{10})$

Hey guys! Ever stumbled upon a math problem that looks like a jumbled mess and wondered, "What is this thing?" Well, today we're diving deep into one of those, specifically simplifying the algebraic expression $(\sqrt{12}+\sqrt{6})(\sqrt{6}-\sqrt{10})$. This might seem intimidating at first glance, but trust me, with a few tricks up our sleeve, we'll make it look super neat and tidy. We're going to break down this expression step-by-step, ensuring we understand each move we make. This isn't just about getting the right answer; it's about understanding the why behind it. We'll be using properties of square roots and a little bit of algebraic manipulation to get there. So, grab your favorite drink, settle in, and let's untangle this mathematical puzzle together. By the end of this, you'll feel more confident tackling similar problems and maybe even impress your friends with your newfound simplification skills. Let's get started!

Understanding the Basics: Square Roots and Expansion

Alright, let's start by looking at the expression itself: $(\sqrt{12}+\sqrt{6})(\sqrt{6}-\sqrt{10})$. The first thing we need to do is simplify any square roots that can be simplified. Remember, a square root can be simplified if the number inside it has a perfect square factor. For instance, $\sqrt{12}$ can be written as $\sqrt{4 \times 3}$, and since 4 is a perfect square ($2^2$), we can pull the 2 out, making it $2\sqrt{3}$. This is a crucial first step because it makes the rest of the calculations much cleaner. Now, let's look at the other terms. $\sqrt{6}$ is already in its simplest form because 6 has no perfect square factors other than 1. Similarly, $\sqrt{10}$ cannot be simplified further since 10 ($2 \times 5$) doesn't have any perfect square factors. So, our expression becomes $(2\sqrt{3}+\sqrt{6})(\sqrt{6}-\sqrt{10})$.

Next up, we need to expand this expression. Think of it like multiplying two binomials (those expressions with two terms). We'll use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last), although in this case, it's just applying the distributive property twice. We multiply each term in the first set of parentheses by each term in the second set of parentheses.

• First: $(2\sqrt{3}) \times (\sqrt{6}) = 2\sqrt{3 \times 6} = 2\sqrt{18}$. We can simplify $\sqrt{18}$ further because $18 = 9 \times 2$, and 9 is a perfect square ($3^2$). So, $2\sqrt{18} = 2 \times 3\sqrt{2} = 6\sqrt{2}$.
• Outer: $(2\sqrt{3}) \times (-\sqrt{10}) = -2\sqrt{3 \times 10} = -2\sqrt{30}$. This one can't be simplified further.
• Inner: $(\sqrt{6}) \times (\sqrt{6}) = \sqrt{6 \times 6} = \sqrt{36}$. And $\sqrt{36}$ is just 6!
• Last: $(\sqrt{6}) \times (-\sqrt{10}) = -\sqrt{6 \times 10} = -\sqrt{60}$. Now, let's simplify $\sqrt{60}$. We look for perfect square factors. $60 = 4 \times 15$, and 4 is a perfect square ($2^2$). So, $-\sqrt{60} = -2\sqrt{15}$.

Putting it all together, our expanded expression is $6\sqrt{2} - 2\sqrt{30} + 6 - 2\sqrt{15}$. Doesn't that look a lot more manageable? We've successfully navigated the initial expansion and simplification, setting ourselves up for the final answer.

Combining Like Terms and Final Simplification

So, after all that expansion and simplification, we arrived at the expression: $6\sqrt{2} - 2\sqrt{30} + 6 - 2\sqrt{15}$. The next crucial step in simplifying any algebraic expression is to combine any like terms. Guys, this is where we tidy everything up. Like terms are terms that have the exact same variable part raised to the exact same power. In our case, we're looking for terms with the same radical part. Let's examine what we have:

• $6\sqrt{2}$: This term has a $\sqrt{2}$.
• $-2\sqrt{30}$: This term has a $\sqrt{30}$.
• $6$: This is a constant term (no radical).
• $-2\sqrt{15}$: This term has a $\sqrt{15}$.

Looking closely, we can see that none of these radical terms ($\sqrt{2}$, $\sqrt{30}$, $\sqrt{15}$) are alike. They all have different numbers inside the square root. This means we cannot combine any of the radical terms with each other. We also can't combine any of the radical terms with the constant term, which is just 6.

Therefore, the expression $6\sqrt{2} - 2\sqrt{30} + 6 - 2\sqrt{15}$ is already in its simplest form. We just need to present it in a standard order. Typically, we write the constant term first, followed by the radical terms in some order (often by the number under the radical, from smallest to largest, or alphabetically if they were variables). In this case, ordering the radicals by the number under the root would give us $6 + 6\sqrt{2} - 2\sqrt{15} - 2\sqrt{30}$. However, the options provided might have a different order. Let's compare our result with the given options:

A. $3 \sqrt{2}-\sqrt{22}+2 \sqrt{3}-4$ B. $6 \sqrt{3}-6 \sqrt{5}$ C. $2 \sqrt{3}+6-2 \sqrt{15}$ D. $6 \sqrt{2}-2 \sqrt{30}+6-2 \sqrt{15}$

When we compare our simplified expression, $6\sqrt{2} - 2\sqrt{30} + 6 - 2\sqrt{15}$, directly to the options, we see a perfect match with Option D. The order of terms is slightly different ($6\sqrt{2} - 2\sqrt{30} + 6 - 2\sqrt{15}$ versus $6 \sqrt{2}-2 \sqrt{30}+6-2 \sqrt{15}$), but the terms themselves are identical. This confirms our calculation.

Conclusion: The Power of Simplification

So, there you have it, folks! We've successfully taken a seemingly complex expression, $(\sqrt{12}+\sqrt{6})(\sqrt{6}-\sqrt{10})$, and simplified it down to $6 \sqrt{2}-2 \sqrt{30}+6-2 \sqrt{15}$. The key takeaways here are the importance of simplifying individual square roots first and then applying the distributive property (or FOIL method) to expand the expression. Remember to always look for perfect square factors within the radicals to simplify them. After expansion, the final step is always to combine like terms. In this particular problem, there were no like terms to combine among the radicals, but it's a vital step to keep in mind for other problems.

This process of simplification is not just about making math problems easier; it's about revealing the underlying structure and elegance of mathematical expressions. By mastering these techniques, you gain a deeper understanding of algebraic manipulation and build a stronger foundation for tackling more advanced mathematical concepts. Whether you're preparing for exams, working on homework, or just curious about math, being able to simplify expressions like this is a superpower. It shows you can break down complexity into manageable parts and arrive at a clear, concise answer. Keep practicing, keep exploring, and don't be afraid to dive into those tricky-looking problems. The satisfaction of simplifying them is totally worth it! So, the answer to what is the following product? $(\sqrt{12}+\sqrt{6})(\sqrt{6}-\sqrt{10})$ is indeed D. $6 \sqrt{2}-2 \sqrt{30}+6-2 \sqrt{15}$. Great job working through this with me!