Simplifying Radical Expressions: A Step-by-Step Guide

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Hey guys! Ever stumbled upon a radical expression that looks like a total mess? Don't worry, we've all been there. Radical expressions can seem intimidating at first, but with a few simple techniques, you can simplify them like a pro. Today, we're going to break down a specific example step-by-step. Let's dive into simplifying the expression: $2 \sqrt{27}+\sqrt{12}-3 \sqrt{3}-2 \sqrt{12}$. By the end of this guide, you'll not only know the answer but also understand the underlying principles that make it all click. So grab your pencils, and let's get started!

Breaking Down the Problem

So, the expression we need to simplify is $2 \sqrt{27}+\sqrt{12}-3 \sqrt{3}-2 \sqrt{12}$. The key to simplifying radical expressions lies in identifying perfect square factors within the radicals. Perfect squares are numbers that result from squaring an integer (e.g., 4, 9, 16, 25, etc.). Our goal is to rewrite the numbers under the square roots as products of perfect squares and other factors. This allows us to take the square root of the perfect square and pull it outside the radical, simplifying the expression.

First, let's look at $2 \sqrt{27}$. We need to find the largest perfect square that divides 27. That perfect square is 9 because $27 = 9 \times 3$. So we can rewrite $2 \sqrt{27}$ as $2 \sqrt{9 \times 3}$. According to the properties of square roots, this becomes $2 \times \sqrt{9} \times \sqrt{3}$, which simplifies to $2 \times 3 \times \sqrt{3} = 6 \sqrt{3}$.

Next, let's tackle $\sqrt{12}$. The largest perfect square that divides 12 is 4, because $12 = 4 \times 3$. Thus, we rewrite $\sqrt{12}$ as $\sqrt{4 \times 3}$. This simplifies to $\sqrt{4} \times \sqrt{3} = 2 \sqrt{3}$.

Now we have all the pieces ready to substitute back into the original expression. This preparatory work of finding perfect square factors is extremely important. It sets the stage for combining like terms, which is the next step. By breaking down each term individually, we prevent mistakes and keep things organized. Understanding the perfect square factors and how they interact with square roots is key to mastering simplification.

Simplifying Each Term

Okay, let's simplify each term in the expression $2 \sqrt{27}+\sqrt{12}-3 \sqrt{3}-2 \sqrt{12}$ individually. We've already done most of the hard work in the previous section, so this should be a breeze.

Starting with the first term, $2 \sqrt{27}$, we found that it simplifies to $6 \sqrt{3}$. Remember, we identified that 27 is $9 \times 3$, so $2 \sqrt{27} = 2 \sqrt{9 \times 3} = 2 \times 3 \sqrt{3} = 6 \sqrt{3}$.

Next, we simplify $\sqrt{12}$. We determined that 12 is $4 \times 3$, so $\sqrt{12} = \sqrt{4 \times 3} = 2 \sqrt{3}$.

The third term, $-3 \sqrt{3}$, is already in its simplest form since 3 has no perfect square factors other than 1. So we just keep it as $-3 \sqrt{3}$.

Finally, we simplify $-2 \sqrt{12}$. We already know that $\sqrt{12} = 2 \sqrt{3}$, so $-2 \sqrt{12} = -2 \times 2 \sqrt{3} = -4 \sqrt{3}$.

Now that we've simplified each term, we can rewrite the original expression as $6 \sqrt{3} + 2 \sqrt{3} - 3 \sqrt{3} - 4 \sqrt{3}$. Simplifying each term individually allows us to confidently combine them in the next step. Knowing how to manipulate these terms independently is crucial, and it makes the whole process much more manageable.

Combining Like Terms

Alright, now comes the fun part: combining like terms! After simplifying each radical term individually, we now have the expression $6 \sqrt{3} + 2 \sqrt{3} - 3 \sqrt{3} - 4 \sqrt{3}$. Notice that each term has the same radical part, which is $\sqrt{3}$. This means they are like terms and we can combine them by simply adding and subtracting their coefficients.

To combine these like terms, we add and subtract the coefficients: $6 + 2 - 3 - 4$. Doing the addition and subtraction, we get $6 + 2 = 8$, then $8 - 3 = 5$, and finally $5 - 4 = 1$. So the coefficient of $\sqrt{3}$ is 1.

Therefore, the simplified expression is $1 \sqrt{3}$, which is simply $\sqrt{3}$. This means that $2 \sqrt{27}+\sqrt{12}-3 \sqrt{3}-2 \sqrt{12}$ simplifies to $\sqrt{3}$. Isn't that neat? By breaking down each term and then combining them, we arrived at a concise and elegant solution. This ability to recognize and combine like terms is crucial not just in simplifying radical expressions, but also in a wide range of algebraic problems.

Final Answer and Wrap Up

So, after all that simplification, we've found that $2 \sqrt{27}+\sqrt{12}-3 \sqrt{3}-2 \sqrt{12} = \sqrt{3}$. Therefore, the answer is A. $\sqrt{3}$.

To recap, here's what we did:

  1. Identified perfect square factors: We broke down each radical term to find perfect square factors.
  2. Simplified each term: We simplified each term by taking the square root of the perfect square factors.
  3. Combined like terms: We combined the simplified terms by adding and subtracting their coefficients.

By following these steps, you can simplify almost any radical expression. Remember, the key is to take it one step at a time and stay organized. And now, you have the skills to simplify radical expressions like a true mathlete. Keep practicing, and you'll become even more confident in your abilities.

Simplifying radical expressions might seem tough, but with practice, you'll get the hang of it! Keep up the great work, and you'll be simplifying complex expressions in no time. Happy calculating!