Simplifying $(4x)^{1/2} + 3(x)^{1/2}$ A Step-by-Step Solution

In the realm of mathematics, simplifying expressions is a fundamental skill. Often, we encounter expressions involving radicals, and it's crucial to understand how to manipulate them correctly. This article delves into the process of simplifying a specific radical expression, providing a detailed explanation to enhance your understanding of this concept. Let's consider the expression: $(4x)^{1/2} + 3(x)^{1/2}$. Our goal is to simplify this expression and determine the correct answer from the given options.

Understanding the Basics of Radicals

Before we dive into the solution, let's establish a solid foundation by reviewing the basics of radicals and their properties. Radicals, often represented by the square root symbol (√), indicate a root of a number. The expression $(x)^{1/2}$ is equivalent to √x, representing the square root of x. Similarly, $(x)^{1/n}$ represents the nth root of x. Understanding this equivalence is crucial for simplifying radical expressions effectively. When working with radicals, several key properties come into play. One important property is the product rule of radicals, which states that √(ab) = √a ⋅ √b for non-negative numbers a and b. This rule allows us to break down radicals into simpler components, making simplification easier. Another essential property is the ability to combine like radicals. Like radicals are those that have the same radicand (the expression under the radical sign). For instance, 2√x and 3√x are like radicals, while 2√x and 2√y are not. Like radicals can be combined by adding or subtracting their coefficients, similar to combining like terms in algebraic expressions. With these fundamental concepts in mind, we can approach the simplification of our given expression with confidence.

Breaking Down the Expression

The given expression is $(4x)^{1/2} + 3(x)^{1/2}$. The first step in simplifying this expression is to address the term $(4x)^{1/2}$. Recognizing that the exponent 1/2 represents a square root, we can rewrite this term as √(4x). Now, we can apply the product rule of radicals, which states that √(ab) = √a ⋅ √b. In our case, a = 4 and b = x, so we can rewrite √(4x) as √4 ⋅ √x. The square root of 4 is 2, so we have 2√x. This simplification is a critical step as it allows us to combine like terms later on. Now, let's turn our attention to the second term in the expression, which is $3(x)^{1/2}$. This term can be rewritten as 3√x, which is already in its simplest form. Notice that both terms, 2√x and 3√x, now have the same radicand, which is x. This means they are like radicals, and we can combine them. Combining like radicals is similar to combining like terms in algebra. We simply add or subtract the coefficients of the radicals while keeping the radicand the same. In our case, the coefficients are 2 and 3, so we add them together. This gives us (2 + 3)√x, which simplifies to 5√x. Therefore, the simplified form of the original expression is 5√x. This process of breaking down the expression and simplifying individual terms is essential for accurately solving the problem.

Combining Like Radicals

Having simplified $(4x)^{1/2}$ to 2√x and recognizing that $3(x)^{1/2}$ is equivalent to 3√x, we are now at a pivotal point in our simplification journey. The expression we are working with is now 2√x + 3√x. The key to simplifying this further lies in recognizing that 2√x and 3√x are like radicals. Like radicals share the same radicand, which in this case is 'x'. This shared radicand allows us to combine these terms in a straightforward manner, much like we combine like terms in algebraic expressions. To combine like radicals, we focus on the coefficients, which are the numbers multiplying the radical term. In our expression, the coefficients are 2 and 3. We simply add these coefficients together while keeping the radical term (√x) the same. So, 2√x + 3√x becomes (2 + 3)√x. This is a fundamental step in simplifying radical expressions, as it consolidates multiple terms into a single, more manageable term. Performing the addition, we get 5√x. This is the simplified form of the expression, where we have successfully combined the like radicals. Understanding the concept of like radicals and how to combine them is crucial for simplifying more complex expressions involving radicals. This step not only simplifies the expression but also brings us closer to identifying the correct answer from the given options.

Identifying the Correct Answer

After simplifying the expression $(4x)^{1/2} + 3(x)^{1/2}$, we arrived at the simplified form 5√x. Now, our task is to identify which of the given options matches this simplified expression. The options are:

A. $3(5x)^{1/2}$ B. $(13x)^{1/2}$ C. $4(5x)^{1/2}$ D. $5(x)^{1/2}$ E. $(7x)^{1/2}$

Let's examine each option closely. Option A, $3(5x)^{1/2}$, can be rewritten as 3√(5x), which is not equivalent to 5√x. Option B, $(13x)^{1/2}$, can be rewritten as √(13x), which is also not equivalent to 5√x. Option C, $4(5x)^{1/2}$, can be rewritten as 4√(5x), which is again not equivalent to 5√x. Option D, $5(x)^{1/2}$, can be rewritten as 5√x, which is exactly the simplified form we obtained. Therefore, Option D is the correct answer. Option E, $(7x)^{1/2}$, can be rewritten as √(7x), which is not equivalent to 5√x. By systematically comparing our simplified expression with each option, we can confidently identify the correct answer. This process highlights the importance of accurate simplification and careful comparison when solving mathematical problems. It also demonstrates how a step-by-step approach can lead to a clear and correct solution.

Conclusion

In conclusion, simplifying radical expressions involves a series of steps, from understanding the basic properties of radicals to combining like terms. In this specific case, we successfully simplified the expression $(4x)^{1/2} + 3(x)^{1/2}$ to 5√x, which corresponds to option D, $5(x)^{1/2}$. This process involved breaking down the expression, applying the product rule of radicals, and combining like radicals. By mastering these techniques, you can confidently tackle a wide range of radical simplification problems. Remember, practice is key to improving your skills in mathematics, so continue to work through various examples to solidify your understanding. The ability to simplify radical expressions is not only essential for academic success but also for various real-world applications in fields such as physics, engineering, and computer science. So, keep practicing, and you'll become proficient in simplifying even the most complex radical expressions.