Simplifying Radical Expressions: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the world of simplifying radical expressions. Specifically, we're going to break down how to simplify the expression $(-2 imes \sqrt{20k}) imes (5 imes \sqrt{8k^3})$. Don't worry if it looks a bit intimidating at first – we'll go through it step by step, making sure you understand every move. Our goal is to make this process clear, concise, and easy to follow. By the end of this guide, you'll be able to confidently tackle similar problems. So, let's get started!

Understanding the Basics: Radicals and Simplification

Before we jump into the problem, let's quickly review some fundamentals. A radical expression is any expression that contains a radical symbol, also known as a square root symbol ($\sqrt{ }$). Simplifying a radical expression means rewriting it in a form where the radicand (the number or expression under the radical) has no perfect square factors (other than 1), and there are no radicals in the denominator. This process often involves factoring and using the properties of radicals to rewrite the expression. For example, $\sqrt{9}$ simplifies to 3 because 9 is a perfect square. Our expression, $(-2\sqrt{20k})(5\sqrt{8k^3})$, requires us to use these concepts along with some algebraic manipulation.

To effectively simplify radical expressions, you need to understand the properties of radicals. One crucial property is the product rule of radicals, which states that $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$. This rule allows us to combine or separate radicals, which is essential for simplifying. We'll also use the fact that $\sqrt{a^2} = a$ for any non-negative number a. Keeping these rules in mind, let’s move forward to solve the given expression. The simplification process usually involves identifying perfect squares within the radicand and extracting their square roots. This reduces the complexity of the expression and makes it easier to work with. For instance, if you have $\sqrt{12}$, you can break it down as $\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$. This process of factorization and simplification is at the heart of what we are doing.

Step-by-Step Simplification of $(-2\sqrt{20k})(5\sqrt{8k^3})$

Let’s start simplifying the expression: $(-2\sqrt{20k})(5\sqrt{8k^3})$. We'll do this step by step to ensure clarity. Follow along closely to grasp each move.

1. Multiply the coefficients: The first thing we'll do is multiply the numbers outside the square roots (the coefficients). We have -2 and 5. So, $-2 \times 5 = -10$. Our expression now looks like: $-10 \times \sqrt{20k} \times \sqrt{8k^3}$.

2. Combine the radicals: Next, use the product rule of radicals to combine the two square roots into one. This means multiplying the radicands (the expressions under the square root signs) together. So, $20k \times 8k^3 = 160k^4$. Our expression becomes: $-10 \times \sqrt{160k^4}$.

3. Simplify the radicand: Now, let's simplify the term inside the square root. We have $\sqrt{160k^4}$. Break down 160 into its prime factors to find perfect squares. We can write 160 as $16 \times 10$. Also, $k^4$ is a perfect square because $k^4 = (k^2)^2$. Thus, we have $\sqrt{16 \times 10 \times k^4}$.

4. Extract the square roots: Now, extract the square roots of the perfect squares. The square root of 16 is 4, and the square root of $k^4$ is $k^2$. Therefore, we have $4k^2\sqrt{10}$.

5. Final simplification: Bring the extracted terms outside the square root and multiply them by the coefficient we found earlier (-10). So, we have $-10 \times 4k^2 \times \sqrt{10}$. This simplifies to $-40k^2\sqrt{10}$.

So, the simplified form of $(-2\sqrt{20k})(5\sqrt{8k^3})$ is $-40k^2\sqrt{10}$.

Choosing the Right Answer

Now, let's see which of the provided options matches our simplified answer, $-40k^2\sqrt{10}$.

A. $-10k^4\sqrt{16}$ B. $-10k^2\sqrt{16}$ C. $-40k^4\sqrt{10}$ D. $-40k^2\sqrt{10}$

Looking at the options, we can see that option D, $-40k^2\sqrt{10}$, is the correct answer. This matches the result we obtained through our step-by-step simplification. This process emphasizes the importance of carefully applying the rules of radicals and simplifying each component of the expression correctly. The key is to break down the problem into manageable steps, making sure to extract perfect squares and combine terms appropriately. Practicing these types of problems will help you become more proficient in simplifying radical expressions.

Tips for Success

Here are some helpful tips to keep in mind when simplifying radical expressions:

• Know your perfect squares: Memorizing the first few perfect squares (4, 9, 16, 25, 36, etc.) will speed up the process of identifying them within the radicand.
• Break down the radicand: Always try to factor the radicand into its prime factors to make it easier to find perfect squares.
• Simplify coefficients first: Multiply the coefficients (the numbers outside the square roots) before simplifying the radicals.
• Combine like terms: If you have multiple radical terms, combine the like terms after simplification.
• Double-check your work: Always go back and review your steps to avoid any calculation errors. It's easy to make a small mistake, so a quick review can save you from a wrong answer.
• Practice regularly: The more you practice, the more comfortable and confident you'll become with simplifying radical expressions. Work through various examples to solidify your understanding.

By following these tips, you can greatly improve your ability to simplify radical expressions. Remember that consistency and attention to detail are key to mastering this skill. Don't be afraid to try different examples and seek help when needed. Mathematics is all about practice and understanding the underlying concepts.

Conclusion: Mastering Radical Simplification

In this guide, we've walked through the step-by-step process of simplifying radical expressions, focusing on the example of $(-2\sqrt{20k})(5\sqrt{8k^3})$. We’ve reviewed the basics, applied the properties of radicals, and arrived at the simplified answer. Remember, the core of simplifying radical expressions lies in understanding the product rule of radicals, identifying and extracting perfect squares, and carefully combining terms. Keep practicing, and you'll become more confident in tackling these types of problems. The ability to simplify radicals is a fundamental skill in algebra and is essential for more advanced mathematical concepts. Always remember to break down the problem into smaller, manageable steps. By consistently applying these techniques, you'll find that simplifying radical expressions becomes much easier and more intuitive. Keep exploring and practicing to build a strong foundation in mathematics!

For more detailed explanations and examples, check out this Khan Academy.