Simplifying $6\[\sqrt{3}(\sqrt{5}-\sqrt{3})\]$

Hey guys! Let's dive into simplifying the expression $6\sqrt{3}(\sqrt{5}-\sqrt{3})$. This is a great exercise in working with radicals and applying the distributive property. We'll break it down into easy-to-follow steps so you can master this type of problem. Ready? Let's get started!

Understanding the Basics: Radicals and the Distributive Property

Before we jump into the simplification, let's refresh our memory on the key concepts involved. First up, radicals. A radical, denoted by the symbol $\sqrt{ }$, represents the root of a number. In our case, we're dealing with square roots, which means we're looking for a number that, when multiplied by itself, equals the number under the radical sign. For example, $\sqrt{9} = 3$ because $3 * 3 = 9$. Easy peasy, right?

Next, we have the distributive property. This property states that multiplying a number by a sum or difference inside parentheses is the same as multiplying the number by each term inside the parentheses separately and then adding or subtracting the results. Mathematically, it looks like this: $a(b - c) = ab - ac$. This is the key to solving our problem.

Now, let's look at the given expression again: $6\sqrt{3}(\sqrt{5}-\sqrt{3})$. Here, $6\sqrt{3}$ is the term outside the parentheses, and $(\sqrt{5}-\sqrt{3})$ is the expression inside. We'll use the distributive property to multiply $6\sqrt{3}$ by both $\sqrt{5}$ and $-\sqrt{3}$. This means we'll perform two multiplications: $6\sqrt{3} * \sqrt{5}$ and $6\sqrt{3} * -\sqrt{3}$.

When multiplying radicals, remember that $\sqrt{a} * \sqrt{b} = \sqrt{a*b}$. This is a crucial rule to keep in mind. Also, if a term outside the radical is multiplied by a radical, it stays outside. For instance, if you have something like $2 * \sqrt{5}$, then the 2 remains outside the radical, giving you $2\sqrt{5}$. So, as you see, it's pretty straightforward, and with these basic concepts, you're all set to tackle the simplification!

Also, it's important to keep in mind that the goal of simplifying radical expressions is to get the expression in its simplest form, which means:

• No perfect square factors are inside the radical (other than 1).
• No radicals are in the denominator (we rationalize if necessary).
• All like terms are combined.

Step-by-Step Simplification: Breaking Down the Expression

Alright, let's break down the simplification process step-by-step. This is where the rubber meets the road, so pay close attention. We will be using the distributive property, we’ll start by multiplying $6\sqrt{3}$ by each term inside the parentheses, which are $\sqrt{5}$ and $-\sqrt{3}$.

• Step 1: Distribute $6\sqrt{3}$ First, multiply $6\sqrt{3}$ by $\sqrt{5}$: $6\sqrt{3} * \sqrt{5} = 6 * (\sqrt{3} * \sqrt{5})$. Now, we use the rule $\sqrt{a} * \sqrt{b} = \sqrt{a*b}$. So, $\sqrt{3} * \sqrt{5} = \sqrt{3*5} = \sqrt{15}$. Thus, $6\sqrt{3} * \sqrt{5} = 6\sqrt{15}$.

  Next, multiply $6\sqrt{3}$ by $-\sqrt{3}$: $6\sqrt{3} * -\sqrt{3} = 6 * (\sqrt{3} * -\sqrt{3})$. Again, $\sqrt{3} * \sqrt{3} = \sqrt{3*3} = \sqrt{9} = 3$. So, $6\sqrt{3} * -\sqrt{3} = 6 * -3 = -18$.

• Step 2: Combine the Results Now that we've distributed $6\sqrt{3}$, we have two terms: $6\sqrt{15}$ and $-18$. Combine these terms: $6\sqrt{15} - 18$.

• Step 3: Check for Further Simplification Let’s take a look at $6\sqrt{15} - 18$ to see if we can simplify further. Can we simplify $\sqrt{15}$? The factors of 15 are 1, 3, 5, and 15. None of these factors (other than 1) are perfect squares. So, $\sqrt{15}$ cannot be simplified further. Also, there are no like terms to combine, so we leave it as is.

• Step 4: Final Answer Therefore, the simplified form of $6\sqrt{3}(\sqrt{5}-\sqrt{3})$ is $6\sqrt{15} - 18$. Congrats, you've done it! We've successfully simplified the expression using the distributive property and the rules of radicals. Wasn't that fun?

Common Mistakes and How to Avoid Them

Alright, let's talk about some common pitfalls people encounter when simplifying expressions like these. Knowing these mistakes can help you avoid them and nail the problem every time.

• Incorrect Distribution: The most frequent mistake is not distributing correctly. Remember to multiply the term outside the parentheses by every term inside the parentheses. Sometimes people forget to multiply by the second term, or they only multiply the numbers and not the radicals, leading to an incorrect answer. Always double-check that you've multiplied by both terms.

• Incorrect Radical Multiplication: Another mistake is in multiplying the radicals. Be sure to use the rule $\sqrt{a} * \sqrt{b} = \sqrt{a*b}$. Sometimes people might get confused and try to add the numbers under the radicals instead of multiplying them. Always ensure you are following the correct rules for radical operations.

• Forgetting to Simplify Further: After the initial distribution and multiplication of radicals, always check if you can simplify the resulting radicals further. For example, if you end up with $\sqrt{12}$, remember that $12 = 4 * 3$, and $\sqrt{4} = 2$, so you can simplify this to $2\sqrt{3}$. Failing to simplify the radical completely is a common error. Always look for perfect square factors under the radical sign.

• Not Combining Like Terms: After simplifying the radicals, if you have terms that can be combined (like terms), make sure you combine them. In our example, we didn't have any like terms, but in other problems, you might. Failing to combine like terms will leave your answer not fully simplified.

• Misunderstanding the Order of Operations: Make sure you follow the correct order of operations (PEMDAS/BODMAS). Parentheses/Brackets come first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally, Addition and Subtraction (from left to right). This ensures you perform the operations in the right order.

By being aware of these common mistakes, you can significantly improve your accuracy and confidence in simplifying radical expressions. Always double-check your work, and don't be afraid to redo the steps if you're not sure. Practice makes perfect, and with practice, these mistakes will become less frequent!

Further Practice and Resources

Want to become a radical simplification guru? Awesome! Here are some suggestions and resources to help you further hone your skills. Remember, the more you practice, the better you will become.

• Practice Problems: The key is to practice! Find a bunch of practice problems online or in your textbook. Work through them step by step, and don’t look at the answers until you've tried the problems yourself. Websites like Khan Academy, Mathway, and Purplemath offer tons of practice problems and step-by-step solutions to check your work.

• Worksheets: Download worksheets with a variety of radical expressions. Worksheets provide a structured way to practice and are often organized by difficulty level. You can find many free worksheets online by searching for “radical simplification worksheets.”

• Online Tutorials: YouTube is a treasure trove of math tutorials. Search for videos on simplifying radical expressions and the distributive property. Many excellent educators explain the concepts clearly, and seeing different examples can really help.

• Textbook Examples: Work through the examples in your math textbook. Textbooks often provide detailed explanations and practice problems with solutions. Make sure to understand the worked examples before trying the practice problems.

• Study Groups: Studying with a friend or a group can be helpful. You can discuss problems, explain concepts to each other, and learn from each other's approaches. Teaching someone else is a fantastic way to solidify your own understanding!

• Khan Academy: Khan Academy is an amazing resource that offers free lessons and practice exercises on simplifying radicals and other math topics. The platform provides personalized learning, so you can work at your own pace and focus on areas where you need more practice.

By following these resources and consistently practicing, you'll be simplifying radical expressions like a pro in no time. Keep up the great work, and remember, mathematics is all about practice and understanding. You got this!