Simplifying Radical Expressions An Equivalent Form Guide

Navigating the world of mathematics often involves simplifying complex expressions, and when it comes to radicals and exponents, a systematic approach is essential. In this comprehensive guide, we will delve into the intricacies of simplifying the expression $2 \sqrt{25 y^8} \cdot 4 \sqrt{6 y^3}$, assuming that $y > 0$. Our mission is to transform this expression into its most simplified equivalent form, carefully dissecting each step to ensure clarity and understanding. Join us as we embark on this mathematical journey, meticulously unraveling the expression and providing insights that will empower you to tackle similar challenges with confidence.

Deconstructing the Expression: A Step-by-Step Approach

At first glance, the expression $2 \sqrt{25 y^8} \cdot 4 \sqrt{6 y^3}$ may appear daunting. However, by methodically breaking it down into smaller, manageable components, we can demystify its complexity. Our primary goal is to simplify each radical term individually before combining them. This approach not only streamlines the process but also minimizes the chances of errors. We will begin by addressing the first radical, $\sqrt{25 y^8}$, and then move on to the second radical, $\sqrt{6 y^3}$, ensuring that each step is crystal clear.

Simplifying the First Radical: Unveiling the Square Root of $25y^8$

The first step in our simplification journey involves tackling the radical $\sqrt{25 y^8}$. This term encapsulates both a numerical coefficient and a variable raised to a power, both of which can be simplified independently. To simplify the numerical coefficient, we need to identify the square root of 25. Since 25 is a perfect square (5 * 5 = 25), its square root is simply 5. Now, let's turn our attention to the variable component, $y^8$. When taking the square root of a variable raised to an even power, we divide the exponent by 2. In this case, the square root of $y^8$ is $y^4$ (8 / 2 = 4). Therefore, the simplified form of $\sqrt{25 y^8}$ is $5y^4$. This foundational simplification is a crucial step in our overall quest to find the equivalent expression.

Untangling the Second Radical: Simplifying $\sqrt{6y^3}$

Our next challenge is to simplify the second radical, $\sqrt{6 y^3}$. This radical presents a slightly different scenario as neither the coefficient 6 nor the variable term $y^3$ are perfect squares. However, we can still simplify this radical by identifying any perfect square factors within the radicand (the expression under the radical). Let's begin with the coefficient 6. The prime factorization of 6 is 2 * 3, neither of which are perfect squares. Therefore, the coefficient 6 cannot be simplified further. Now, let's consider the variable term $y^3$. We can rewrite $y^3$ as $y^2 \cdot y$. Here, $y^2$ is a perfect square, and its square root is simply y. The remaining factor, y, stays under the radical. Putting it all together, the simplified form of $\sqrt{6 y^3}$ is $y \sqrt{6 y}$. This step demonstrates the importance of recognizing and extracting perfect square factors from radicals.

Merging Simplified Terms: Multiplying and Consolidating

With both radicals now simplified, we are ready to combine the terms and achieve our final simplified expression. Recall that the original expression was $2 \sqrt{25 y^8} \cdot 4 \sqrt{6 y^3}$. We have simplified $\sqrt{25 y^8}$ to $5y^4$ and $\sqrt{6 y^3}$ to $y \sqrt{6 y}$. Substituting these simplified terms back into the original expression, we get $2 (5y^4) \cdot 4 (y \sqrt{6 y})$. The next step is to multiply the coefficients together: 2 * 5 * 4 = 40. Then, we multiply the variable terms: $y^4 \cdot y = y^5$. Finally, we bring along the remaining radical term, $\sqrt{6 y}$. This gives us the expression $40 y^5 \sqrt{6 y}$, which is the simplified equivalent of the original expression.

Identifying the Equivalent Expression: Choosing the Correct Answer

Having meticulously simplified the expression $2 \sqrt{25 y^8} \cdot 4 \sqrt{6 y^3}$ to $40 y^5 \sqrt{6 y}$, we are now equipped to identify the correct answer from the given options. Let's revisit the options:

A. $40 y^5 \sqrt{6 y}$ B. $14 y^5 \sqrt{6 y^2}$ C. $17 y^2 \sqrt{6 y^5}$ D. $8 y^2 \sqrt{150 y}$

By comparing our simplified expression, $40 y^5 \sqrt{6 y}$, with the options provided, it becomes clear that option A, $40 y^5 \sqrt{6 y}$, is the correct answer. This process highlights the importance of accurate simplification and careful comparison to ensure the correct solution is identified.

In Conclusion: Mastering Radical Simplification

Simplifying radical expressions is a fundamental skill in mathematics, and by understanding the underlying principles and employing a systematic approach, even complex expressions can be tamed. In this guide, we have meticulously dissected the expression $2 \sqrt{25 y^8} \cdot 4 \sqrt{6 y^3}$, demonstrating the step-by-step process of simplifying radicals, combining terms, and identifying the equivalent expression. We began by simplifying each radical individually, extracting perfect square factors where possible. Then, we multiplied the simplified terms together, combining coefficients and variables. Finally, we compared our simplified expression with the given options, confidently selecting the correct answer. Remember, practice is key to mastering these skills. By consistently applying these techniques, you will build confidence and proficiency in simplifying radical expressions.

Which expression is equivalent to $2 \sqrt{25 y^8} \cdot 4 \sqrt{6 y^3}$, if $y>0$?

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