Simplifying Radicals How To Solve 5√12 - 2√27

Hey there, math enthusiasts! Let's dive into simplifying radical expressions. Radicals might seem intimidating at first, but with a bit of practice, you'll be able to handle them like a pro. In this article, we'll break down the expression $5\sqrt{12} - 2\sqrt{27}$ and find its equivalent form. We'll go through the steps together, so you'll understand exactly how to simplify these types of expressions. So, grab your pencils, and let's get started!

Understanding the Problem

Our main task is to simplify the expression $5\sqrt{12} - 2\sqrt{27}$. To do this, we need to understand the properties of square roots and how to manipulate them. Remember, the goal is to get the expression into its simplest form, which usually means reducing the numbers inside the square roots as much as possible. Simplifying radicals involves finding perfect square factors within the radicands (the numbers inside the square root symbol) and taking their square roots. Let’s break down each term separately and then combine them.

Breaking Down $5\sqrt{12}$

Let's focus on simplifying $5\sqrt{12}$ first. To simplify this, we need to find the largest perfect square that divides 12. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25). The factors of 12 are 1, 2, 3, 4, 6, and 12. Among these, 4 is the largest perfect square. So, we can rewrite 12 as $4 \times 3$. Now, we have:

$5\sqrt{12} = 5\sqrt{4 \times 3}$

Using the property of square roots, $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can separate the square root:

$5\sqrt{4 \times 3} = 5 \times \sqrt{4} \times \sqrt{3}$

Since $\sqrt{4} = 2$, we can substitute that in:

$5 \times 2 \times \sqrt{3} = 10\sqrt{3}$

So, $5\sqrt{12}$ simplifies to $10\sqrt{3}$. We’ve taken the first step in simplifying our overall expression. Next, we'll tackle the second term, $2\sqrt{27}$.

Simplifying $2\sqrt{27}$

Now, let's simplify $2\sqrt{27}$. We follow a similar process as before: find the largest perfect square that divides 27. The factors of 27 are 1, 3, 9, and 27. Among these, 9 is the largest perfect square. We can rewrite 27 as $9 \times 3$. Therefore,

$2\sqrt{27} = 2\sqrt{9 \times 3}$

Again, we use the property $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to separate the square root:

$2\sqrt{9 \times 3} = 2 \times \sqrt{9} \times \sqrt{3}$

Since $\sqrt{9} = 3$, we substitute that in:

$2 \times 3 \times \sqrt{3} = 6\sqrt{3}$

So, $2\sqrt{27}$ simplifies to $6\sqrt{3}$. We've now simplified both individual terms. The next step is to combine them.

Combining the Simplified Terms

After simplifying each term separately, we found that $5\sqrt{12} = 10\sqrt{3}$ and $2\sqrt{27} = 6\sqrt{3}$. Now, we can substitute these simplified forms back into our original expression:

$5\sqrt{12} - 2\sqrt{27} = 10\sqrt{3} - 6\sqrt{3}$

Now, we have like terms, meaning they have the same radical part ($\sqrt{3}$). We can combine them just like we combine algebraic terms:

$10\sqrt{3} - 6\sqrt{3} = (10 - 6)\sqrt{3}$

Subtracting the coefficients gives us:

$(10 - 6)\sqrt{3} = 4\sqrt{3}$

So, the simplified form of the expression is $4\sqrt{3}$. This is our final answer.

Choosing the Correct Option

Now that we've simplified the expression $5\sqrt{12} - 2\sqrt{27}$ to $4\sqrt{3}$, we can look at the given options and choose the correct one. The options were:

a) $4\sqrt{3}$ b) $\frac{5}{15}$ c) $2\sqrt{3}$ d) $\frac{3}{15}$

Our simplified expression matches option (a), which is $4\sqrt{3}$. So, the correct answer is indeed $4\sqrt{3}$. We’ve successfully simplified the expression and found its equivalent form.

Common Mistakes to Avoid

When simplifying radicals, it's easy to make a few common mistakes. Let's go over these so you can avoid them. One frequent mistake is not finding the largest perfect square factor. For example, when simplifying $\sqrt{12}$, you might identify 4 as a perfect square factor but overlook that 4 is the largest perfect square factor. Missing the largest factor means you'll have to simplify further down the line, so always aim for the largest one right away.

Another mistake is incorrectly applying the distributive property or combining terms that are not like terms. Remember, you can only combine terms with the same radical part. For instance, $3\sqrt{2} + 4\sqrt{3}$ cannot be simplified further because $\sqrt{2}$ and $\sqrt{3}$ are different. Also, make sure you correctly apply the properties of square roots, like $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, to avoid errors.

Lastly, forgetting to simplify the coefficients after simplifying the radicals is another common oversight. After you’ve simplified the radicals, double-check if the coefficients can be further simplified or combined. Avoiding these mistakes will help you simplify radical expressions accurately and efficiently.

Practice Problems

To really nail down simplifying radicals, practice is key! Here are a few problems for you to try on your own. Work through them step by step, and remember to look for the largest perfect square factors.

1. Simplify $3\sqrt{8} - \sqrt{18}$
2. Simplify $2\sqrt{20} + 3\sqrt{45}$
3. Simplify $4\sqrt{24} - \sqrt{54}$

Working through these problems will help you build confidence and skill in simplifying radical expressions. Remember, take your time, break down each term, and don't forget to double-check your work. Practice makes perfect, so keep at it!

Conclusion

Simplifying radical expressions might seem tricky at first, but as we've seen, it's all about breaking down the problem into smaller, manageable steps. By finding perfect square factors, using the properties of square roots, and combining like terms, you can simplify even complex expressions. Remember, always look for the largest perfect square factor to make your work easier. And, of course, practice is essential to mastering these skills.

In this article, we walked through simplifying $5\sqrt{12} - 2\sqrt{27}$, and we found that it simplifies to $4\sqrt{3}$. By understanding the process and avoiding common mistakes, you can tackle similar problems with confidence. So keep practicing, and you’ll become a pro at simplifying radicals in no time! Keep up the great work, mathletes!