Simplify Radical Expressions: $6y ext{sqrt}(a) + 7y ext{sqrt}(a)$

Hey guys, today we're diving into a pretty straightforward, but super important, math concept: simplifying algebraic expressions, specifically those involving radicals. So, you've got this expression: $6 y \sqrt{a}+7 y \sqrt{a}$. The challenge is to find the sum of these two terms. Now, for all you math whizzes out there, this might seem like a piece of cake, but for those who are just getting their feet wet with algebra and radicals, it can be a bit daunting. Don't worry, though! We're going to break it down step-by-step, making sure everyone can follow along and feel confident. Think of it like combining like terms in regular algebra, but with an added radical twist. The key here is to recognize that both terms have the same variable part, $y \sqrt{a}$. This is what allows us to combine them. It's like having 6 apples and adding 7 more apples – you end up with 13 apples, right? Well, here, instead of apples, we have $y \sqrt{a}$ units. So, when we add $6 y \sqrt{a}$ and $7 y \sqrt{a}$, we're essentially adding the coefficients, which are the numbers multiplying the variable part. The coefficients are 6 and 7. Adding them gives us $6 + 7 = 13$. The variable part, $y \sqrt{a}$, remains the same because it's common to both terms. Therefore, the sum is $13 y \sqrt{a}$. This is a fundamental skill in algebra, and mastering it will open doors to tackling more complex problems down the line. We'll also touch upon why the other options are incorrect, helping you understand the nuances of simplifying these kinds of expressions. So, grab your notebooks, get comfortable, and let's get this math party started!

Understanding the Basics of Combining Like Terms with Radicals

Alright, let's get down to business with our problem: $6 y \sqrt{a}+7 y \sqrt{a}$. Before we jump into the solution, it's crucial to understand the core principle at play here, which is combining like terms. In algebra, like terms are terms that have the exact same variables raised to the exact same powers. Think of the variables as labels. If the labels are identical, you can combine the numerical coefficients. For example, in $3x + 5x$, both terms have the variable $x$. So, we can combine their coefficients, 3 and 5, to get $8x$. Similarly, in $2y^2 + 9y^2$, both terms have $y^2$, allowing us to combine the coefficients 2 and 9 to get $11y^2$. Now, when we introduce radicals, like $\sqrt{a}$, the same principle applies. A term is considered 'like' another term if it has the same variable part, including the radical part, raised to the same power. In our expression, $6 y \sqrt{a}+7 y \sqrt{a}$, let's examine each term. The first term is $6 y \sqrt{a}$. Its variable part is $y \sqrt{a}$. The second term is $7 y \sqrt{a}$. Its variable part is also $y \sqrt{a}$. Since both terms share the identical variable part, $y \sqrt{a}$, they are like terms. This means we can add their coefficients. The coefficient of the first term is 6, and the coefficient of the second term is 7. Adding these coefficients gives us $6 + 7 = 13$. The common variable part, $y \sqrt{a}$, remains unchanged. So, the sum of $6 y \sqrt{a}$ and $7 y \sqrt{a}$ is $13 y \sqrt{a}$. It's as simple as that! You're not adding the 'a' inside the radical, nor are you changing the power of 'y'. You're just combining the quantities of the identical $y \sqrt{a}$ units. This concept is fundamental and forms the basis for simplifying much more complex algebraic expressions involving radicals. Keep this in mind as we move forward, because this understanding is the key to unlocking the solution.

Solving the Expression: Step-by-Step

Let's walk through the solution for $6 y \sqrt{a}+7 y \sqrt{a}$ in a clear, methodical way. Our goal is to find the sum of these two terms.

Step 1: Identify the terms. We have two terms: $6 y \sqrt{a}$ and $7 y \sqrt{a}$.

Step 2: Check if the terms are 'like terms'. To be like terms, they must have the exact same variable part. In this case, the variable part of the first term is $y \sqrt{a}$, and the variable part of the second term is also $y \sqrt{a}$. Since both variable parts are identical, these are indeed like terms.

Step 3: Combine the coefficients. Since the terms are like terms, we can add their numerical coefficients. The coefficient of the first term is 6, and the coefficient of the second term is 7.

So, we perform the addition: $6 + 7 = 13$.

Step 4: Write the final simplified expression. The sum is the new coefficient (13) followed by the common variable part ($y \sqrt{a}$).

Therefore, the sum of $6 y \sqrt{a}+7 y \sqrt{a}$ is $\mathbf{13 y \sqrt{a}}$.

It's important to note that we only combine the coefficients when the variable parts are identical. If, for instance, we had $6 y \sqrt{a} + 7 y \sqrt{b}$, we couldn't combine them because the radical parts are different ($\sqrt{a}$ vs $\sqrt{b}$). Similarly, if we had $6 y \sqrt{a} + 7 y^2 \sqrt{a}$, we couldn't combine them because the powers of 'y' are different (y vs $y^2$). The simplicity of our original problem lies in the fact that both terms perfectly match, allowing for a direct combination of their coefficients. This step-by-step process ensures accuracy and builds confidence in handling such algebraic manipulations. Remember, the key is always to identify those 'like terms' first!

Analyzing the Answer Choices

Now that we've found the correct sum, let's take a moment to look at the provided answer choices and understand why our answer is correct and the others are not. This helps solidify our understanding and prevents common mistakes.

Our expression is: $6 y \sqrt{a}+7 y \sqrt{a}$ Our calculated sum is: $\mathbf{13 y \sqrt{a}}$

Let's examine the options:

• A. $13 y^2 \sqrt{a}$: This answer is incorrect because it changes the power of the variable 'y'. When combining like terms, the variable part remains the same. We are adding $6(y\sqrt{a})$ and $7(y\sqrt{a})$. We are not multiplying the 'y' terms together, which would result in $y^2$. The operation here is addition, not multiplication of the variable parts. Thus, the 'y' should remain as 'y', not $y^2$.

• B. $13ay$: This answer is incorrect because it incorrectly modifies the radical part. The original terms have $\sqrt{a}$. Combining them should result in a term that still includes $\sqrt{a}$. This option replaces $\sqrt{a}$ with 'a', which is a completely different term. You can't just substitute the radical sign with the radicand itself when adding.

• C. $13 y \sqrt{2 a}$: This answer is incorrect because it alters the radical in a way that isn't justified by the addition. Adding $6 y \sqrt{a}$ and $7 y \sqrt{a}$ means you have 13 units of $y \sqrt{a}$. You are not adding the numbers inside the square roots. If you were multiplying terms like $\sqrt{a} \times \sqrt{2}$, you might get $\sqrt{2a}$, but this is an addition problem. The quantity under the radical, 'a', remains unchanged in its form within the sum.

• D. $13 y \sqrt{a}$: This answer is correct! It perfectly matches our calculation. We identified that $6 y \sqrt{a}$ and $7 y \sqrt{a}$ are like terms because they both have the identical variable part $y \sqrt{a}$. We then added their coefficients, $6 + 7 = 13$. The resulting sum is simply the new coefficient (13) multiplied by the common variable part ($y \sqrt{a}$), which gives us $13 y \sqrt{a}$.

By dissecting each option, we reinforce why D is the only valid answer. It adheres strictly to the rules of combining like terms with radicals, ensuring that only the coefficients are added while the variable and radical components remain unchanged. Keep this analysis in mind for future problems!

Why This Matters: The Foundation of Algebra

So, why do we spend time on seemingly simple problems like finding the sum of $6 y \sqrt{a}+7 y \sqrt{a}$? Because, guys, this is the bedrock of more advanced mathematics. Understanding how to combine like terms, especially with radicals, is not just about getting the right answer on a quiz; it's about building a solid foundation for everything else you'll encounter in algebra and beyond. Think about it: later on, you'll be dealing with expressions that have multiple terms, different types of radicals, exponents, and a whole host of other mathematical elements. If you haven't truly grasped the fundamental concept of identifying and combining like terms, those more complex problems will feel like trying to build a skyscraper on quicksand – it's just not going to hold up.

Mastering this skill means you can simplify complex equations, which is essential for solving them. Imagine a complicated physics problem or an engineering calculation. Often, the first step involves simplifying a messy algebraic expression into something manageable. That simplification relies heavily on combining like terms. Furthermore, understanding radicals is crucial in many fields, including geometry (think Pythagorean theorem!), calculus, and statistics. The ability to manipulate expressions like $6 y \sqrt{a}+7 y \sqrt{a}$ demonstrates your proficiency with mathematical operations and your logical thinking process. It shows you can see patterns, identify similarities (like those 'like terms'), and perform operations based on established rules. This process trains your brain to think systematically and analytically. So, the next time you're faced with an expression like this, remember its importance. It's not just an exercise; it's a vital step in your mathematical journey, equipping you with the tools needed to tackle bigger challenges and appreciate the elegance of mathematical simplification. Keep practicing, stay curious, and you'll be amazed at what you can achieve!