Simplifying 2√62 + 3√20 - 2√28 + 3√80 A Step-by-Step Guide

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Hey guys! Let's break down this math problem together. We've got the expression 2√62 + 3√20 - 2√28 + 3√80, and our goal is to simplify it. This involves understanding how to work with square roots and identifying any perfect square factors within the radicals. Trust me, it's not as scary as it looks! We'll take it step by step, making sure everyone's on board. So, grab your thinking caps, and let's dive into the world of simplifying square root expressions!

Breaking Down the Radicals

To effectively simplify the expression, 2√62 + 3√20 - 2√28 + 3√80, we need to first focus on breaking down each radical individually. The core idea here is to identify if there are any perfect square factors hiding inside the numbers under the square root signs. Perfect squares are numbers that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). By extracting these perfect squares, we can simplify the radicals and make the overall expression much easier to handle. Let's start by examining each term:

  1. 2√62: In this case, 62 can be factored as 2 * 31. Unfortunately, neither 2 nor 31 is a perfect square. Therefore, √62 cannot be simplified further. This means the term 2√62 remains as is for now.

  2. 3√20: Here, 20 can be factored into 4 * 5. Notice that 4 is a perfect square (2 * 2 = 4). We can rewrite √20 as √(4 * 5). Using the property of square roots that √(a * b) = √a * √b, we can further break this down into √4 * √5. Since √4 is 2, the term simplifies to 2√5. Now, multiplying this by the coefficient 3, we get 3 * 2√5, which equals 6√5. So, 3√20 simplifies to 6√5.

  3. -2√28: For this term, 28 can be factored into 4 * 7. Again, 4 is a perfect square. We rewrite √28 as √(4 * 7), which can be separated into √4 * √7. The square root of 4 is 2, so we have 2√7. Multiplying by the coefficient -2, we get -2 * 2√7, which equals -4√7. Thus, -2√28 simplifies to -4√7.

  4. 3√80: Lastly, 80 can be factored into 16 * 5. The number 16 is a perfect square (4 * 4 = 16). We rewrite √80 as √(16 * 5), which becomes √16 * √5. The square root of 16 is 4, giving us 4√5. Multiplying by the coefficient 3, we get 3 * 4√5, which simplifies to 12√5. Therefore, 3√80 simplifies to 12√5.

By breaking down each radical in this manner, we transform the original expression into a form where we can more easily identify like terms and combine them. This initial step is crucial for simplifying complex expressions involving radicals. Now that we have simplified each radical individually, we can move on to the next phase: combining like terms.

Combining Like Terms

Now that we've simplified each radical in the expression 2√62 + 3√20 - 2√28 + 3√80, we've transformed it into 2√62 + 6√5 - 4√7 + 12√5. The next step is to identify and combine the like terms. In the context of radicals, like terms are those that have the same number under the square root sign. Think of the radical part (like √5 or √7) as a variable – you can only add or subtract terms that have the same variable.

Looking at our expression, we can see that we have a couple of terms with √5: 6√5 and 12√5. These are like terms, and we can combine them by simply adding their coefficients (the numbers in front of the radical). So, 6√5 + 12√5 becomes (6 + 12)√5, which equals 18√5. This is similar to combining 6x and 12x to get 18x in algebra.

The other terms, 2√62 and -4√7, do not have any like terms in the expression. This means they cannot be combined with any other terms and will remain as they are in our simplified expression. The key here is that √62 and √7 represent different values, just like x and y in algebra are different variables.

So, after combining the like terms, we take the simplified term 18√5 and put it together with the remaining terms. Our expression now looks like this: 2√62 - 4√7 + 18√5. We've successfully grouped and combined the terms that could be simplified, leaving us with an expression that is much cleaner and easier to understand than the original. This process of identifying and combining like terms is a fundamental step in simplifying any algebraic or radical expression, and it's a skill that comes in handy in many areas of mathematics.

Final Simplified Expression

After breaking down the radicals and combining like terms, we've arrived at the final simplified expression. Starting with 2√62 + 3√20 - 2√28 + 3√80, we simplified it step by step. We identified perfect square factors within the radicals, extracted them, and then combined the terms with the same radical. Remember, the initial simplification gave us 2√62 + 6√5 - 4√7 + 12√5. Then, we combined the like terms 6√5 and 12√5 to get 18√5.

As a result of these simplifications, we have the expression 2√62 - 4√7 + 18√5. This is the most simplified form of the original expression. There are no more like terms to combine, and each radical is in its simplest form. It's important to present the answer in a clear and organized manner, so anyone looking at your work can easily understand the final result.

This final expression showcases the power of simplification in mathematics. What started as a seemingly complex expression with multiple terms and different radicals has been transformed into a much more manageable and understandable form. This is not only important for getting the correct answer but also for developing a deeper understanding of the underlying mathematical concepts. Remember, guys, simplifying expressions is a crucial skill in algebra and beyond, so mastering these techniques will definitely pay off in the long run!

Conclusion

So, to wrap things up, we successfully simplified the expression 2√62 + 3√20 - 2√28 + 3√80. We walked through each step, from breaking down the radicals to combining like terms, and ended up with the simplified form 2√62 - 4√7 + 18√5. This journey highlights the importance of understanding the properties of square roots and how to manipulate them effectively.

The key takeaways here are:

  1. Breaking down radicals: Always look for perfect square factors within the numbers under the square root sign. This is the first step to simplifying any radical expression.
  2. Combining like terms: Just like in algebra, you can only add or subtract terms that have the same radical part. Think of the radical as a variable.
  3. Step-by-step approach: Simplifying complex expressions becomes much easier when you break it down into smaller, manageable steps. This not only makes the process less daunting but also reduces the chances of making errors.

By following these steps and practicing regularly, you'll become more confident and proficient in simplifying radical expressions. Remember, math is like building blocks – each concept builds upon the previous one. Mastering these fundamental skills will set you up for success in more advanced topics. Keep practicing, keep exploring, and most importantly, keep enjoying the process of learning!