Simplifying Square Roots: A Step-by-Step Guide

Hey guys! Today, we're diving into a math problem that might seem a little intimidating at first glance: Simplifying 2√62 + 3√20 - 2√28 + 3√80. Don't worry, though; we'll break it down step by step to make it super easy to understand. This kind of problem is all about simplifying square roots by finding perfect square factors. So, grab your calculators (or not, if you're feeling brave!), and let's get started. This detailed guide will not only help you solve this specific problem but also equip you with the skills to tackle similar problems with confidence. Understanding how to simplify square roots is a fundamental concept in algebra and is super useful for higher-level math.

Step 1: Prime Factorization and Identifying Perfect Squares

Alright, the first thing we need to do is to simplify each of the square roots in our expression. The key here is prime factorization. This means breaking down each number under the square root into its prime factors – prime numbers that, when multiplied together, give you the original number. Let's start with each term:

• 2√62: First, factorize 62 into its prime factors. 62 = 2 x 31. Neither 2 nor 31 is a perfect square, and there are no repeated factors. So, this term cannot be simplified further. It remains as 2√62.
• 3√20: Next, factorize 20. 20 = 2 x 2 x 5 or 2² x 5. Notice that we have a pair of 2s (2²). This indicates a perfect square. We can rewrite 3√20 as 3√(2² x 5). Taking the square root of 2² gives us 2. So, 3√20 simplifies to 3 x 2√5, which equals 6√5.
• -2√28: Let's factorize 28. 28 = 2 x 2 x 7 or 2² x 7. Again, we see a pair of 2s. This simplifies as -2√(2² x 7) which equals -2 x 2√7, resulting in -4√7.
• 3√80: Lastly, factorize 80. 80 = 2 x 2 x 2 x 2 x 5 or 2⁴ x 5 or (2² x 2²) x 5. We can rewrite 3√80 as 3√((2² x 2²) x 5). Simplifying this will be 3 x 2 x 2√5, which equals 12√5. By simplifying the radicals, we're essentially making the numbers under the square root as small as possible, which will make the final addition and subtraction much easier. Finding the perfect squares is critical for simplification. Remember, a perfect square is a number that results from squaring an integer (e.g., 4, 9, 16, 25). This step of prime factorization is where the magic happens; it's all about spotting those pairs or groups of numbers that allow you to simplify the square root.

Summary of Step 1

After prime factorization, our expression now looks like this: 2√62 + 6√5 - 4√7 + 12√5. We went through each term to break it down into its simplest radical form. This process is the foundation for simplifying the entire expression.

Step 2: Grouping Like Terms

Now that we've simplified the individual square roots, the next step involves grouping like terms. Like terms are terms that have the same radical (the number under the square root). Looking at our simplified expression: 2√62 + 6√5 - 4√7 + 12√5, we can see two terms containing √5. We can combine 6√5 and 12√5. The other terms, 2√62 and -4√7, do not have any like terms, so they remain as they are.

• Combining 6√5 and 12√5, we simply add their coefficients (the numbers in front of the square root): 6 + 12 = 18. So, 6√5 + 12√5 becomes 18√5.

Updated Expression

Our expression now looks like this: 2√62 + 18√5 - 4√7. Notice that we were able to simplify two terms with the same radical, making the expression more manageable. This step is straightforward once you have identified the like terms correctly. Always double-check to make sure you're only combining terms with identical radicals. This ensures that your simplification is accurate. Remember, we cannot combine terms with different radicals, such as √5 and √7. They are not like terms and must stay separate. The ability to group like terms is crucial for simplifying expressions and solving equations effectively. It is important for organization.

Step 3: Final Simplification

In the final step, we will combine the terms we found, if possible. However, looking at our expression: 2√62 + 18√5 - 4√7, there are no more like terms to combine. This means that we have reached the simplest form of our expression. The terms have different radicals (√62, √5, and √7), so we cannot simplify any further. The coefficients and the radicals are now in their simplest forms, and we cannot perform any additional operations to reduce this further. The simplification process is complete! Sometimes, after simplifying, you might find that there are still no like terms to combine, as is the case here. This means you've done your best, and the expression is as simple as it can get. Always ensure you have performed all possible simplifications at each stage of the process.

Final Answer

The simplified form of the original expression, 2√62 + 3√20 - 2√28 + 3√80, is 2√62 + 18√5 - 4√7. We’ve taken a complex-looking problem and broken it down into easy, manageable steps. By understanding prime factorization, identifying perfect squares, and grouping like terms, you can tackle similar problems with confidence. This method can be used for a range of radical expressions. Great job!

Conclusion: Mastering Square Root Simplification

So, guys, we've successfully simplified the expression! It's important to practice these steps with different problems to master the technique. The main takeaways from this process include prime factorization, identifying perfect squares, and combining like terms. Remember, the more you practice, the easier it becomes. Square root simplification is an essential skill in algebra and is used in many other areas of mathematics. By understanding and practicing these steps, you’ll be well-equipped to handle more complex mathematical problems. Keep practicing, and you’ll find that simplifying square roots becomes second nature. Good luck!