Simplifying (2-5√(6)) * 3√(2) A Step-by-Step Guide

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Guys, let's dive into simplifying this mathematical expression: (2-5\sqrt{6}) ullet 3\sqrt{2}. It might look a little intimidating at first glance, but don't worry! We're going to break it down step by step, making it super easy to understand. Our goal here is to clarify each step, ensuring that everyone, whether you're a math whiz or just getting started, can follow along and grasp the concepts. So, grab your pencils and let’s get started!

Understanding the Basics of Expression Simplification

Before we jump into the nitty-gritty of this particular expression, let’s quickly recap the basic principles of expression simplification. In mathematics, simplifying an expression means rewriting it in its most basic and compact form. This usually involves performing all possible arithmetic operations, combining like terms, and getting rid of any unnecessary clutter. When we talk about simplifying expressions, we're essentially trying to make them easier to work with and understand. Think of it like decluttering your room – you want to get rid of anything that’s not essential and organize what's left in a neat way.

The Distributive Property: Our Key Tool

One of the most important tools in our simplification toolkit is the distributive property. This property allows us to multiply a single term by a sum or difference inside parentheses. In simple terms, it states that $a(b + c) = ab + ac$. This might seem like a straightforward concept, but it's incredibly powerful when dealing with more complex expressions, especially those involving radicals like our expression today. The distributive property is the key to unlocking many mathematical problems, and we’ll be using it extensively in this simplification process. So, keep this principle in mind as we move forward.

Working with Radicals: A Quick Refresher

Radicals, like the square root of 6 ($\sqrt{6}$) in our expression, can sometimes seem a bit mysterious. But they’re really just numbers expressed in a different form. A radical represents a root of a number; for example, $\sqrt{4}$ is 2 because 2 squared is 4. When simplifying expressions with radicals, we often look for ways to combine them or to simplify them individually. One important rule to remember is that \sqrt{a} ullet \sqrt{b} = \sqrt{ab}, which can help us combine radicals under a single root. Understanding how to manipulate radicals is crucial for simplifying expressions like ours, and we’ll be sure to highlight these techniques as we go along.

Step-by-Step Simplification of (2-5\sqrt{6}) ullet 3\sqrt{2}

Okay, let’s get down to business! We’re going to simplify the expression (2-5\sqrt{6}) ullet 3\sqrt{2} step by step. This way, you can see exactly how we arrive at the final answer, making the process clear and easy to follow.

Step 1: Applying the Distributive Property

The first thing we need to do is apply the distributive property. Remember, this means multiplying the term outside the parentheses ($3\sqrt{2}$) by each term inside the parentheses ($2$ and $-5\sqrt{6}$). So, we have:

3\sqrt{2} ullet 2 - 3\sqrt{2} ullet 5\sqrt{6}

This step is crucial because it allows us to break down the original expression into smaller, more manageable pieces. By applying the distributive property correctly, we've set the stage for the next steps in our simplification journey.

Step 2: Multiplying the Terms

Now, let's multiply the terms we've created. First, we multiply $3\sqrt{2}$ by 2, which gives us $6\sqrt{2}$. Next, we multiply $3\sqrt{2}$ by $5\sqrt{6}$. To do this, we multiply the coefficients (3 and 5) and the radicals ($\sqrt{2}$ and $\sqrt{6}$) separately:

3 ullet 5 = 15

\sqrt{2} ullet \sqrt{6} = \sqrt{2 ullet 6} = \sqrt{12}

So, 3\sqrt{2} ullet 5\sqrt{6} = 15\sqrt{12}. Now, our expression looks like this:

$6\sqrt{2} - 15\sqrt{12}$

This step involves careful multiplication, ensuring we keep track of both the coefficients and the radicals. By multiplying the terms correctly, we’re one step closer to our simplified expression.

Step 3: Simplifying the Radical $\sqrt{12}$

You might notice that $\sqrt{12}$ can be simplified further. To do this, we need to find the largest perfect square that divides 12. A perfect square is a number that is the square of an integer (e.g., 4, 9, 16, etc.). In this case, the largest perfect square that divides 12 is 4. So, we can rewrite $\sqrt{12}$ as:

\sqrt{12} = \sqrt{4 ullet 3} = \sqrt{4} ullet \sqrt{3} = 2\sqrt{3}

Now, we substitute this back into our expression:

$6\sqrt{2} - 15(2\sqrt{3})$

Simplifying radicals is a key part of the process, and by breaking down $\sqrt{12}$, we’ve made our expression even simpler.

Step 4: Final Simplification

Finally, let’s simplify the expression by multiplying 15 by $2\sqrt{3}$:

$15(2\sqrt{3}) = 30\sqrt{3}$

So, our expression now looks like this:

$6\sqrt{2} - 30\sqrt{3}$

And that’s it! We’ve simplified the expression as much as possible. There are no more like terms to combine, and the radicals are in their simplest form. Our final answer is:

$6\sqrt{2} - 30\sqrt{3}$

This final step ties everything together, and we can see the result of our hard work. By final simplification, we’ve reached the most compact and understandable form of the original expression.

Common Mistakes to Avoid When Simplifying Expressions

Simplifying expressions can be tricky, and it’s easy to make mistakes if you’re not careful. Let's go over some common pitfalls to help you avoid them. Knowing what to watch out for can save you a lot of headaches and ensure you get the correct answer every time.

Mistake 1: Forgetting the Distributive Property

One of the most common mistakes is forgetting to distribute the term outside the parentheses to every term inside. Remember, the distributive property is your friend, and it applies to each term within the parentheses. For example, in our original expression, we had to multiply $3\sqrt{2}$ by both 2 and $-5\sqrt{6}$. If you miss one, your entire simplification will be off. So, always double-check that you’ve applied the distributive property fully.

Mistake 2: Incorrectly Multiplying Radicals

Multiplying radicals can be a bit confusing if you’re not careful. Remember, you multiply the coefficients (the numbers in front of the radicals) separately from the radicals themselves. And don’t forget the rule: \sqrt{a} ullet \sqrt{b} = \sqrt{ab}. A common mistake is to multiply the coefficients with the radicals, which is a big no-no. Always keep the coefficients and radicals separate during multiplication, and you’ll be on the right track.

Mistake 3: Not Simplifying Radicals Completely

Another frequent error is not simplifying radicals as much as possible. Always look for perfect square factors within the radical and simplify them. For example, we simplified $\sqrt{12}$ to $2\sqrt{3}$. If you leave radicals unsimplified, you’re not getting the expression to its simplest form. So, make sure to simplify radicals completely to get the correct final answer.

Mistake 4: Combining Unlike Terms

This is a classic mistake in algebra. You can only combine like terms, which means terms with the same variable and exponent. In our case, we had $6\sqrt{2}$ and $30\sqrt{3}$. These are unlike terms because they have different radicals ($\sqrt{2}$ and $\sqrt{3}$), so we can’t combine them. Trying to combine unlike terms will lead to an incorrect simplification. Always ensure you’re only adding or subtracting terms that are truly alike.

Mistake 5: Sign Errors

Sign errors are surprisingly common, especially when dealing with negative numbers. Always pay close attention to the signs in front of each term and make sure you’re applying the correct operations. A simple sign error can throw off the entire simplification. So, watch out for sign errors and double-check your work to avoid these sneaky mistakes.

Practice Problems for Mastering Expression Simplification

Okay, guys, now that we’ve gone through the steps and common mistakes, it’s time to put your skills to the test! Practice is the key to mastering expression simplification. The more you practice, the more comfortable and confident you’ll become. Here are a few practice problems for you to try. Work through them step by step, and don’t forget the tips and tricks we discussed earlier.

Problem 1: Simplify (3 + 2\sqrt{5}) ullet 4\sqrt{3}

This problem is similar to the one we just worked through. Start by applying the distributive property, then multiply the terms, simplify the radicals, and combine like terms if possible. Remember to take it one step at a time and double-check your work.

Problem 2: Simplify (7 - \sqrt{2}) ullet 2\sqrt{7}

Another great problem to practice the distributive property and radical simplification. Pay close attention to the signs and make sure you’re simplifying the radicals completely.

Problem 3: Simplify (4\sqrt{3} - 2) ullet 5\sqrt{2}

This problem will help you practice multiplying and simplifying radicals. Remember to keep the coefficients and radicals separate during multiplication and look for perfect square factors within the radicals.

Tips for Practicing

• Work step by step: Break each problem down into smaller, manageable steps. This will make the process less overwhelming and help you catch any mistakes early on.
• Show your work: Write down each step clearly. This will not only help you keep track of your progress but also make it easier to identify any errors.
• Check your answers: After you’ve completed a problem, check your answer against the solution. If you made a mistake, go back and see where you went wrong. Understanding your mistakes is a crucial part of the learning process.
• Practice regularly: The more you practice, the better you’ll become. Try to set aside some time each day or week to work on simplification problems.

By tackling these practice problems, you’ll reinforce your understanding of expression simplification and develop the skills you need to excel in math. So, grab your pencils, dive in, and have fun!

Conclusion: Mastering Expression Simplification

Alright, guys, we’ve reached the end of our journey through expression simplification! We’ve covered everything from the basic principles to step-by-step solutions and common mistakes to avoid. By now, you should have a solid understanding of how to simplify expressions, even those that look intimidating at first glance.

Simplifying expressions is a fundamental skill in mathematics, and it’s essential for success in more advanced topics. Whether you’re solving equations, graphing functions, or tackling calculus problems, the ability to simplify expressions will be invaluable. Remember, the key is to take it one step at a time, apply the rules and properties correctly, and practice, practice, practice!

We started by understanding the basics of expression simplification and the importance of the distributive property. We then walked through a detailed example, simplifying (2-5\sqrt{6}) ullet 3\sqrt{2} step by step. We also highlighted common mistakes to avoid, such as forgetting the distributive property, incorrectly multiplying radicals, and not simplifying radicals completely. Finally, we provided practice problems for you to try, reinforcing your skills and building your confidence.

So, keep practicing, keep learning, and keep simplifying! You’ve got this!