Simplify $(3-7 imes 6)(8+3 imes 6)$ Expression

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the world of mathematics, specifically tackling a problem that might look a bit intimidating at first glance: multiplying and simplifying algebraic expressions involving square roots. Don't worry, though! We're going to break it down step-by-step, making sure you feel super confident by the end. This kind of problem is super common in algebra, and once you get the hang of it, you'll be zipping through them like a pro. We're talking about expressions that look like $(a-b imes c)(d+e imes c)$, where $c$ is a square root. The key here is understanding how to distribute terms correctly and then how to combine like terms. It's all about organization and applying the right rules. So, grab your favorite snack, settle in, and let's get this math party started! We'll be using the distributive property, often called FOIL (First, Outer, Inner, Last), to systematically multiply each term in the first parenthesis by each term in the second parenthesis. Then, we'll simplify any square roots and combine any terms that have the same radical part. It's a solid process that, with a little practice, becomes second nature. We're not just solving this one problem; we're building a foundation for tackling more complex expressions down the line. So, pay close attention to each step, and feel free to pause and rewind if needed. Let's conquer this algebraic beast together!

Understanding the Basics of Square Roots

Before we jump into the main problem, let's quickly refresh our memory on a few key concepts, especially regarding square roots. A square root, denoted by the radical symbol $\sqrt{}$, is the inverse operation of squaring a number. For example, the square root of 9 is 3 because $3^2 = 9$. When we encounter square roots in algebraic expressions, like $\sqrt{6}$ in our problem, it's crucial to remember that they are numbers, just like integers or fractions. We can perform arithmetic operations on them. One of the most important rules for square roots is that the product of square roots is the square root of the product: $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$. This is super handy. For instance, $\sqrt{2} \times \sqrt{3} = \sqrt{2 \times 3} = \sqrt{6}$. Also, if we have a square root of a number that has a perfect square factor, we can simplify it. For example, $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$. In our problem, we have $\sqrt{6}$. Since 6 doesn't have any perfect square factors other than 1 (its factors are 1, 2, 3, and 6), $\sqrt{6}$ is already in its simplest form. This is important because it means we won't be able to simplify it further during our calculations. We'll be treating $\sqrt{6}$ as a single unit, much like a variable, when we combine terms later. So, when you see $\sqrt{6}$, just think of it as a special kind of number that needs careful handling, especially when multiplying or adding/subtracting with other terms. The rules for these operations are consistent, and mastering them is key to simplifying expressions efficiently. The goal is always to reduce the expression to its simplest possible form, which means no more terms can be combined and all square roots are simplified. Understanding these fundamental properties of square roots is the bedrock upon which our entire simplification process will be built. Without this solid foundation, trying to navigate the complexities of multiplying binomials with radicals would be like trying to build a house without blueprints – messy and prone to collapse! So, let's ensure we're all on the same page with these radical rules. Remember, $\sqrt{a} \times \sqrt{a} = a$, which is a direct consequence of $\sqrt{a} = a^{1/2}$ and $a^{1/2} \times a^{1/2} = a^{1/2+1/2} = a^1 = a$. This specific property will be very important when we multiply terms that both contain $\sqrt{6}$. It's the shortcut that simplifies things dramatically.

Applying the Distributive Property (FOIL)

Alright, fam, now let's get to the core of the problem: multiplying the two binomials. Our expression is $(3-7 \sqrt{6})(8+3 \sqrt{6})$. To multiply these, we'll use the distributive property. The most common way to remember this for binomials is the FOIL method: First, Outer, Inner, Last. Let's break down what each letter means in our specific problem:

• F (First): Multiply the first terms in each binomial. That's $3 \times 8$.
• O (Outer): Multiply the outer terms. That's $3 \times (3 \sqrt{6})$.
• I (Inner): Multiply the inner terms. That's $(-7 \sqrt{6}) \times 8$.
• L (Last): Multiply the last terms. That's $(-7 \sqrt{6}) \times (3 \sqrt{6})$.

Let's do the math for each step:

1. First: $3 \times 8 = 24$. Easy peasy!

2. Outer: $3 \times (3 \sqrt{6})$. Here, we multiply the whole numbers together: $3 \times 3 = 9$. So, this term becomes $9 \sqrt{6}$.

3. Inner: $(-7 \sqrt{6}) \times 8$. Again, multiply the whole numbers: $-7 \times 8 = -56$. So, this term is $-56 \sqrt{6}$. Remember the negative sign!

4. Last: $(-7 \sqrt{6}) \times (3 \sqrt{6})$. This one has a few more parts. First, multiply the whole numbers: $-7 \times 3 = -21$. Then, multiply the square roots: $\sqrt{6} \times \sqrt{6}$. As we discussed earlier, $\sqrt{6} \times \sqrt{6} = 6$. So, this whole term becomes $-21 \times 6$.

Now, let's calculate $-21 \times 6$. We know $20 \times 6 = 120$, and $1 \times 6 = 6$. So, $21 \times 6 = 120 + 6 = 126$. Therefore, the last term is $-126$.

So, after applying FOIL, our expression looks like this:

$24 + 9 \sqrt{6} - 56 \sqrt{6} - 126$

This is the result of the multiplication. It's not simplified yet, but we've successfully multiplied the two binomials. The FOIL method systematically ensures that every term in the first binomial is multiplied by every term in the second binomial, preventing any potential misses. It's a structured approach that guarantees completeness. Remember to pay close attention to the signs of each term. A common mistake is misplacing a negative sign during these multiplications. Double-checking each step of the FOIL process, especially the signs and the multiplication of radicals, is highly recommended. When you're first learning, it might be helpful to write out each multiplication separately, just like we did here, to ensure accuracy. Once you've done it a few times, you'll start to see the patterns and can do it more quickly in your head. The goal is to transform the product of two binomials into a sum or difference of terms, which is exactly what we've achieved. The next step, of course, is to simplify this expanded expression by combining like terms, which we'll tackle next. It’s about transforming a complex product into a more manageable sum.

Combining Like Terms for Simplification

We're almost there, guys! Our current expression is $24 + 9 \sqrt{6} - 56 \sqrt{6} - 126$. The next crucial step in simplifying is to combine like terms. In this expression, we have two types of terms: constant terms (numbers without any square roots) and terms with $\sqrt{6}$.

Let's identify them:

• Constant terms: We have $24$ and $-126$.
• Terms with $\sqrt{6}$: We have $9 \sqrt{6}$ and $-56 \sqrt{6}$.

Now, let's combine each group:

1. Combine the constant terms: $24 - 126$. Since $126$ is larger than $24$ and it's negative, the result will be negative. $126 - 24 = 102$. So, $24 - 126 = -102$.

2. Combine the terms with $\sqrt{6}$: $9 \sqrt{6} - 56 \sqrt{6}$. Think of $\sqrt{6}$ as a unit. We're essentially combining $9$ of these units with $-56$ of these units. So, $9 - 56$. This will also be negative. $56 - 9 = 47$. Therefore, $9 \sqrt{6} - 56 \sqrt{6} = -47 \sqrt{6}$.

Now, let's put these combined terms back together. The simplified expression is the sum of our combined constants and our combined radical terms:

$-102 - 47 \sqrt{6}$

And that, my friends, is our final answer! We have successfully multiplied and simplified the expression $(3-7 \sqrt{6})(8+3 \sqrt{6})$. The expression is now in its simplest form because we have no like terms left to combine, and the square root $\sqrt{6}$ cannot be simplified further. It’s always a good idea to do a quick check to ensure no further simplification is possible. For instance, if we had ended up with something like $\sqrt{12}$ in our answer, we would need to go back and simplify that to $2\sqrt{3}$. But here, $\sqrt{6}$ is as simple as it gets. The process of combining like terms is fundamental to simplifying any algebraic expression. It's where we group similar components together to reduce the overall complexity. This involves treating terms with the same variable or radical part as a single entity for the purpose of addition and subtraction. For example, in $5x + 2x - 3y$, we combine $5x$ and $2x$ to get $7x$, leaving the expression as $7x - 3y$. The same logic applies to terms with radicals. The expression $-102 - 47 \sqrt{6}$ is the most concise representation of the original product. We've taken a product of two binomials, expanded it using the distributive property, and then condensed it by combining like terms. This methodical approach ensures accuracy and leads to the simplest form. Keep practicing this, and you'll be a simplification wizard in no time!

Final Answer and Recap

So, to recap the whole journey, we started with the expression $(3-7 \sqrt{6})(8+3 \sqrt{6})$. Our goal was to multiply and simplify it. First, we tackled the multiplication using the FOIL method (First, Outer, Inner, Last). This gave us:

• First: $3 \times 8 = 24$
• Outer: $3 \times 3\sqrt{6} = 9\sqrt{6}$
• Inner: $-7\sqrt{6} \times 8 = -56\sqrt{6}$
• Last: $-7\sqrt{6} \times 3\sqrt{6} = -21 \times 6 = -126$

Putting these together, we got $24 + 9\sqrt{6} - 56\sqrt{6} - 126$.

Next, we moved on to simplifying by combining like terms. We grouped the constant terms and the terms with $\sqrt{6}$:

• Constants: $24 - 126 = -102$
• Radical terms: $9\sqrt{6} - 56\sqrt{6} = -47\sqrt{6}$

Finally, we combined these results to get our final simplified answer: $-102 - 47\sqrt{6}$.

This problem really highlights the importance of careful, step-by-step work in mathematics. Every step, from understanding square root properties to correctly applying the distributive property and then diligently combining like terms, plays a vital role. It's like building with LEGOs – each piece has to be placed correctly for the final structure to be sound. When you encounter similar problems, remember these steps: distribute, multiply, and then combine. Also, always keep an eye out for opportunities to simplify square roots, although in this specific case, $\sqrt{6}$ was already in its simplest form. The ability to confidently multiply and simplify expressions involving radicals is a fundamental skill in algebra and opens the door to understanding more advanced mathematical concepts. Keep practicing, stay curious, and don't be afraid to ask questions! Mathematics is a journey of discovery, and problems like these are just stepping stones. So next time you see an expression like this, you'll know exactly how to handle it. You guys are math whizzes! Keep up the amazing work, and we'll catch you in the next article for more math adventures!