Simplifying Radical Expressions: A Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of simplifying radical expressions. Today, we're going to tackle a problem that involves multiplying radicals. Specifically, we'll be simplifying the expression $2 \/sqrt{5 x^3}\left(-3 \/sqrt{10 x^2}\right)$. This might seem a bit daunting at first, but trust me, with a systematic approach, it's totally manageable. We'll break it down into simple steps, ensuring that you understand the process and can confidently solve similar problems in the future. So, grab your pencils, and let's get started on this mathematical adventure!

Understanding the Basics: Radicals and Their Properties

Before we jump into the problem, let's brush up on some fundamental concepts related to radicals. Radicals, also known as roots, are mathematical expressions that represent the inverse operation of exponentiation. The most common type of radical is the square root, denoted by the symbol $\/sqrt{\quad}$. For example, $\/sqrt{9}$ is the square root of 9, which equals 3 because $3^2 = 9$. Understanding the basic properties of radicals is super important for simplifying expressions. One key property is that the product of two square roots is equal to the square root of the product of the numbers under the radicals. Mathematically, this is expressed as $\/sqrt{a} \cdot \/sqrt{b} = \/sqrt{a \cdot b}$. This property is the cornerstone of our simplification process. We'll also need to remember that $\/sqrt{x^2} = x$, assuming x is non-negative, since we are dealing with square roots. This means that if we encounter a term like $x^2$ under the radical, we can simplify it to x outside the radical. Finally, it’s worth noting the commutative and associative properties of multiplication which allow us to rearrange terms as needed. These basics will guide us in every single step.

Now, let's explore this further. When we deal with radicals involving variables, like $x$ in our problem, we need to consider the constraints on those variables. Generally, we assume that variables under square roots are non-negative to avoid dealing with complex numbers. So, in our case, $x$ must be greater than or equal to zero. This constraint is crucial because it ensures that our solutions are real numbers. Moreover, when simplifying radical expressions, our goal is usually to express them in their simplest form. This means removing any perfect square factors from the radicand (the expression under the radical) and ensuring that there are no radicals in the denominator. To remove perfect square factors, we look for terms like $x^2$ or numerical perfect squares like 4, 9, 16, etc. and simplify them outside the radical. By keeping these basic concepts in mind, we're setting a strong foundation for simplifying complex radical expressions efficiently. So, let’s move on to the actual solution of our problem.

Step-by-Step Solution: Multiplying the Radicals

Alright, let's get down to the nitty-gritty and solve the expression $2 \/sqrt{5 x^3}\left(-3 \/sqrt{10 x^2}\right)$. Here’s a breakdown of the steps to ensure clarity:

Step 1: Multiply the coefficients and the radicands separately.

First, multiply the coefficients (the numbers outside the square roots): $2 \cdot -3 = -6$. Then, multiply the radicands (the expressions inside the square roots): $5x^3 \cdot 10x^2 = 50x^5$. So far, we have $-6 \/sqrt{50x^5}$.

Step 2: Simplify the radical.

Next, simplify the radical $\/sqrt{50x^5}$. We can break down 50 into its prime factors: $50 = 2 \cdot 25 = 2 \cdot 5^2$. Also, we can rewrite $x^5$ as $x^4 \cdot x = (x^2)^2 \cdot x$. Now, rewrite the radical: $\/sqrt{50x^5} = \/sqrt{2 \cdot 5^2 \cdot (x^2)^2 \cdot x}$.

Step 3: Extract perfect squares.

Now, extract the perfect squares from under the radical. We have $5^2$ and $(x^2)^2$. The square root of $5^2$ is 5, and the square root of $(x^2)^2$ is $x^2$. So, we can pull these terms out of the radical: $\/sqrt{2 \cdot 5^2 \cdot (x^2)^2 \cdot x} = 5x^2 \/sqrt{2x}$.

Step 4: Combine all the results.

Finally, multiply the simplified radical by the coefficient we found in Step 1. Remember that we had $-6$ as the coefficient. So, we multiply $-6 \cdot 5x^2 \/sqrt{2x}$. This gives us $-30x^2 \/sqrt{2x}$.

Therefore, the simplified form of $2 \/sqrt{5 x^3}\left(-3 \/sqrt{10 x^2}\right)$ is $-30x^2 \/sqrt{2x}$.

Analyzing the Answer Choices and Conclusion

Now, let's look back at the answer choices provided. We've calculated the simplified expression to be $-30x^2 \/sqrt{2x}$. Let's compare this to the options given in the problem:

A. $-30 \/sqrt{2 x^5}$ B. $-30 x^2 \/sqrt{2 x}$ C. $-12 x^2 \/sqrt{5 x}$ D. $-6 \/sqrt{50 x^5}$

From our calculations, option B. $-30 x^2 \/sqrt{2 x}$ is the correct answer. The other options either have incorrect coefficients or have not been simplified correctly. For instance, options A, C, and D have errors in either the coefficient or the terms within the radical, indicating mistakes in the simplification process. Remember, the key to simplifying radical expressions lies in correctly identifying perfect square factors and applying the properties of radicals step by step. Also, the constraints on the variables (in this case, x >= 0) is important to ensure that the solution is valid within the real number system. Keep practicing and you will get the hang of it, guys!

To wrap it up, simplifying radicals might appear tricky at first, but when you break it down into manageable steps, it becomes quite straightforward. Always remember the fundamental rules – multiply coefficients, multiply radicands, identify and extract perfect squares, and then combine everything. Practicing consistently will definitely boost your confidence and proficiency in handling these types of problems. So keep practicing and never give up; with each problem you solve, you'll become more and more proficient! You got this!