Simplifying Expressions: Adding Radicals

Hey guys! Let's dive into a common math problem: finding the sum of expressions involving radicals. Specifically, we're going to tackle the problem: Find the sum $6y\sqrt{a} + 7y\sqrt{a}$. This is a great example of how to combine terms with radicals, and it's super important for building a solid foundation in algebra. Don't worry, it's not as scary as it looks! We'll break it down step-by-step to make sure everyone understands. Adding radicals is similar to combining like terms in regular algebra. The key thing to remember is that you can only add or subtract radicals if they have the same radicand (the number or expression inside the radical sign) and the same index (the small number indicating the root, which is 2 for a square root). In our case, the radicand is 'a' and the index is 2 (since it's a square root), and they are the same in both terms, which makes our life easier. The process involves treating the radical part ($\sqrt{a}$) as a common factor, and then combining the coefficients (the numbers in front of the radical). This is like saying, "I have 6 apples and 7 apples, how many apples do I have in total?" You simply add the numbers in front and keep the 'apple' part the same. This method is incredibly useful for simplifying algebraic expressions, solving equations, and understanding more complex mathematical concepts. So, let's get started and unravel this problem together! We will explore all the steps and show you how to easily arrive at the correct answer.

Now, let's get down to business and figure out this math problem. We'll show you how simple this really is and give you all the information you need. Understanding the underlying principles of adding radicals is a fundamental skill in algebra and is used extensively in other areas of mathematics. Being able to quickly and accurately solve these types of problems is crucial for success. Now, let's explore this example step by step. We'll first look at the original equation. Then, we will break down each step so that you know exactly how to get to the answer. Keep in mind that understanding this concept will help you solve more complicated equations in the future. We'll make sure to explain everything clearly and simply, so you won't have any trouble following along. So let's get started and find out the answer to this expression.

Step-by-Step Solution

Alright, let's crack this problem! The expression we're dealing with is $6y\sqrt{a} + 7y\sqrt{a}$. The goal is to add these two terms together. Remember, we treat the radical part ($\sqrt{a}$) like a common factor. This means we can combine the coefficients (the numbers in front of the radical) while keeping the radical part the same. It's essentially like saying: "We have 6 of something plus 7 of the same something". To solve this, we will add the coefficients together, in this case, 6 and 7. Thus, we have 6 + 7 = 13. We keep the $y\sqrt{a}$ part the same. Therefore, the simplified expression becomes $13y\sqrt{a}$. This simplification process is a building block for more complex algebraic manipulations. It is crucial to grasp these basic principles. This will help you easily solve more complex problems later on. So, in short, we have to recognize that since both terms have the same radical, we can add their coefficients. We then get our final solution! Now, let's look at the correct answer choice from the options.

So, the answer is D. $13y\sqrt{a}$.

Why Other Options Are Incorrect

Let's quickly go through why the other answer choices are incorrect. This is also important to understand because it reinforces the concepts we are learning and helps us avoid common mistakes.

• A. $13y^2\sqrt{a}$: This answer choice is incorrect because it involves squaring the 'y' term. We don't square 'y' when adding the terms; we only add the coefficients. The radical and the variable 'y' remain the same. This answer would be correct if we were multiplying something, but in this case, we are adding.

• B. 13ay: This option incorrectly combines the 'a' inside the radical with the 'y' outside, and it also removes the radical sign altogether. This changes the entire mathematical operation. Remember, you can only combine terms that have the same radical components. This is not the case here, therefore it is incorrect.

• C. $13y\sqrt{2a}$: This option is incorrect because it changes the value inside the radical. When adding radicals, the value inside the square root must remain the same. This answer suggests a manipulation of the original radical that does not occur when adding terms. Remember, we have to keep the radical component as it is in the original expression.

Understanding why the incorrect options are wrong helps solidify your understanding of adding radicals. Always remember to add the coefficients while keeping the radical and variable parts unchanged. This process is consistent throughout algebra and is a crucial concept to master for future studies in mathematics. Always double-check your work, and you'll be on your way to acing these types of problems!

Conclusion: Mastering Radical Addition

Alright, guys, we did it! We successfully simplified the expression $6y\sqrt{a} + 7y\sqrt{a}$ and found that the sum is $13y\sqrt{a}$. Adding radicals is a fundamental skill in algebra, and now you have a better understanding of how it works. Always remember that you can only add terms with the same radical. Combine the coefficients and keep the radical part the same. Practice makes perfect, so try some similar problems on your own to reinforce your skills. Mastering these basic concepts will set you up for success in more advanced mathematical topics. Keep practicing, stay curious, and you'll become a pro in no time! So, keep up the great work, and don't hesitate to revisit this guide whenever you need a quick refresher. See ya!