Solving (6√6-7√2) * 2√2: A Step-by-Step Guide

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Solving (6√6-7√2) * 2√2: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun math problem that involves simplifying an expression with square roots. Specifically, we're going to break down the expression (6√6 - 7√2) * 2√2 step by step, so you can follow along and understand exactly how to solve it. Whether you're a student tackling homework or just someone who enjoys a good math challenge, this guide is for you. Let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have an expression that involves multiplying a term containing square roots by another square root. The key here is to distribute the 2√2 across the terms inside the parentheses and then simplify. Remember, square roots can be a little tricky, but with a systematic approach, we can easily handle them. So, grab your pencil and paper, and let's get to work!

Breaking Down the Components

Our expression is (6√6 - 7√2) * 2√2. To solve this, we'll use the distributive property, which means we multiply each term inside the parentheses by 2√2. This gives us two separate multiplication problems:

  1. (6√6) * (2√2)
  2. (7√2) * (2√2)

We'll tackle each of these one at a time, simplifying as we go. Remember, when multiplying square roots, you multiply the numbers outside the square root and the numbers inside the square root separately.

Multiplying the Terms

Let's start with the first term: (6√6) * (2√2).

To multiply these, we multiply the coefficients (the numbers outside the square root) and the radicands (the numbers inside the square root). So, we have:

  • 6 * 2 = 12 (multiplying the coefficients)
  • √6 * √2 = √12 (multiplying the radicands)

So, (6√6) * (2√2) = 12√12. But we're not done yet! We can simplify √12 further because 12 has a perfect square factor.

Simplifying the Square Root

To simplify √12, we need to find the largest perfect square that divides 12. In this case, it's 4, since 12 = 4 * 3. So, we can rewrite √12 as √(4 * 3). Using the property of square roots, we can separate this into √4 * √3. Since √4 = 2, we have:

√12 = 2√3

Now, we substitute this back into our expression:

12√12 = 12 * (2√3) = 24√3

So, the first term simplifies to 24√3.

Moving on to the Second Term

Now, let's tackle the second term: (7√2) * (2√2).

Again, we multiply the coefficients and the radicands:

  • 7 * 2 = 14 (multiplying the coefficients)
  • √2 * √2 = √4 (multiplying the radicands)

So, (7√2) * (2√2) = 14√4.

Simplifying the Second Square Root

Simplifying √4 is easy because 4 is a perfect square. √4 = 2. So, we have:

14√4 = 14 * 2 = 28

Thus, the second term simplifies to 28.

Combining the Simplified Terms

Now that we've simplified both terms, we can put them back together. Remember, our original expression was (6√6 - 7√2) * 2√2, which we broke down into (6√6) * (2√2) - (7√2) * (2√2). We found that:

  • (6√6) * (2√2) = 24√3
  • (7√2) * (2√2) = 28

So, our expression now looks like this:

24√3 - 28

Final Answer

Therefore, (6√6 - 7√2) * 2√2 = 24√3 - 28. This is the simplified form of the expression. There are no like terms to combine, so we're done!

Tips for Solving Similar Problems

When you're faced with expressions involving square roots, here are a few tips to keep in mind:

  1. Distribute Properly: Make sure to distribute any terms outside parentheses to each term inside the parentheses.
  2. Multiply Coefficients and Radicands Separately: When multiplying square roots, multiply the numbers outside the square root and the numbers inside the square root separately.
  3. Simplify Square Roots: Always look for perfect square factors within the square root and simplify them.
  4. Combine Like Terms: If there are like terms (terms with the same square root), combine them to simplify further.
  5. Stay Organized: Keep your work organized to avoid mistakes. Write down each step clearly.

Additional Tips

  • Practice Makes Perfect: The more you practice, the better you'll get at simplifying expressions with square roots.
  • Review Perfect Squares: Knowing your perfect squares (1, 4, 9, 16, 25, etc.) will help you quickly simplify square roots.
  • Use Prime Factorization: If you're having trouble finding perfect square factors, use prime factorization to break down the number into its prime factors.
  • Check Your Work: Always double-check your work to make sure you haven't made any mistakes.

Common Mistakes to Avoid

  • Forgetting to Distribute: Make sure to distribute to every term inside the parentheses.
  • Incorrectly Multiplying Square Roots: Remember to multiply coefficients with coefficients and radicands with radicands.
  • Not Simplifying Square Roots: Always simplify square roots to their simplest form.
  • Combining Unlike Terms: Only combine terms that have the same square root.
  • Arithmetic Errors: Be careful with your arithmetic. Double-check your calculations.

Conclusion

So there you have it! We've successfully solved the expression (6√6 - 7√2) * 2√2 and simplified it to 24√3 - 28. I hope this step-by-step guide has been helpful and that you now feel more confident in tackling similar problems. Remember to practice regularly, stay organized, and double-check your work. Keep up the great work, and I'll see you in the next math adventure! You got this, guys!