Equivalent Expression For Sqrt(128x^8y^3z^9) Simplification Guide

Introduction to Simplifying Radicals

When dealing with expressions involving radicals, it's crucial to understand how to simplify them effectively. In this article, we will delve into the process of simplifying the expression $\sqrt{128 x^8 y^3 z^9}$, assuming that $y \geq 0$ and $z \geq 0$. This involves breaking down the expression under the square root into its prime factors and identifying perfect squares. The ability to simplify radicals is fundamental in various areas of mathematics, including algebra, calculus, and beyond. Our focus will be on identifying the correct equivalent expression from a given set of options, emphasizing the steps and reasoning behind the simplification process. Understanding these concepts will enable you to tackle similar problems with confidence and accuracy.

Simplifying radical expressions often involves extracting perfect squares from under the radical sign. This process hinges on the property $\sqrt{a^2} = |a|$, which simplifies to $a$ when $a$ is non-negative, as is the case with our variables $y$ and $z$. Let's start by examining the numerical coefficient, 128. We need to find its prime factorization to identify any perfect square factors. Prime factorization is a critical step because it allows us to see the fundamental building blocks of a number and how they can be rearranged. The prime factorization of 128 is $2^7$, which can be rewritten as $2^6 \cdot 2$. The $2^6$ part is a perfect square because it can be expressed as $(2^3)^2 = 8^2$. This means we can take the square root of $2^6$ and bring it outside the radical. In the context of simplifying algebraic expressions, this approach of breaking down numbers and variables into their prime factors is indispensable for handling more complex radicals.

Now, let's consider the variable parts of the expression: $x^8$, $y^3$, and $z^9$. To simplify these under the square root, we look for even powers, as they represent perfect squares. For $x^8$, since 8 is an even number, we can easily take its square root. The square root of $x^8$ is $x^4$ because $(x^4)^2 = x^8$. For $y^3$, we can rewrite it as $y^2 \cdot y$. The $y^2$ is a perfect square, and its square root is $y$ (since we are given $y \geq 0$). The remaining $y$ will stay under the radical. Similarly, for $z^9$, we can rewrite it as $z^8 \cdot z$. The $z^8$ is a perfect square, and its square root is $z^4$ because $(z^4)^2 = z^8$. The remaining $z$ will stay under the radical. This technique of separating variables with even powers is a cornerstone of simplifying radical expressions, enabling us to reduce complex expressions into more manageable forms. The process not only simplifies the expression but also makes it easier to perform further algebraic manipulations.

By combining these steps, we can rewrite the original expression $\sqrt{128 x^8 y^3 z^9}$ by extracting the square roots of the perfect square factors. This involves taking the square root of $2^6$, $x^8$, $y^2$, and $z^8$, and leaving the remaining factors under the radical. This methodical approach of breaking down the problem into smaller, manageable parts is key to successfully simplifying radicals. The ability to recognize and extract perfect squares allows us to transform complex expressions into simpler, equivalent forms, which is a crucial skill in advanced mathematical studies. The simplified expression will reveal which of the given options is the correct equivalent form. This highlights the importance of a systematic approach in mathematics, where each step builds upon the previous one to reach the solution.

Step-by-Step Simplification

To find the equivalent expression, we will methodically simplify $\sqrt{128 x^8 y^3 z^9}$. The key here is to break down each component under the radical into its prime factors and perfect squares. This approach not only simplifies the radical but also makes it easier to identify terms that can be brought outside the square root. The ability to deconstruct complex terms into simpler components is a fundamental skill in algebra and beyond, allowing for a more intuitive understanding of the mathematical structure involved. By focusing on each part of the expression individually, we can minimize errors and ensure a more accurate simplification process.

First, we address the numerical coefficient, 128. As we discussed earlier, the prime factorization of 128 is $2^7$. This can be rewritten as $2^6 \cdot 2$, where $2^6$ is a perfect square. The square root of $2^6$ is $2^3$, which equals 8. Thus, we have identified a significant component that can be extracted from the radical. Understanding the prime factorization of numbers is crucial because it allows us to identify and isolate perfect squares, which are essential for simplifying radicals. This step-by-step approach ensures that we do not overlook any factors and that the simplification is done correctly. The ability to manipulate numbers in this way is a core skill in mathematics, providing a foundation for more advanced topics.

Next, we consider the variable terms. For $x^8$, since 8 is an even exponent, $x^8$ is a perfect square. The square root of $x^8$ is $x^4$ because $(x^4)^2 = x^8$. For $y^3$, we rewrite it as $y^2 \cdot y$. The $y^2$ is a perfect square, and its square root is $y$ (given $y \geq 0$). The remaining $y$ stays under the radical. Similarly, for $z^9$, we rewrite it as $z^8 \cdot z$. The $z^8$ is a perfect square, and its square root is $z^4$ because $(z^4)^2 = z^8$. The remaining $z$ stays under the radical. This process of handling variables with exponents is a direct application of the properties of exponents and radicals, demonstrating how these concepts intertwine in algebraic manipulations. By treating each variable term separately, we maintain clarity and reduce the chance of making mistakes.

Now, we combine the simplified components. We have extracted 8 from the coefficient, $x^4$ from $x^8$, $y$ from $y^3$, and $z^4$ from $z^9$. The terms remaining under the radical are 2, $y$, and $z$. Therefore, the simplified expression becomes $8 x^4 y z^4 \sqrt{2 y z}$. This step is the culmination of our previous efforts, bringing together all the simplified components into a single, coherent expression. The ability to synthesize these individual parts into a final answer is a key aspect of problem-solving in mathematics. This final expression allows us to easily compare our result with the given options and determine the correct equivalent expression.

Evaluating the Options

After simplifying the original expression, we have $8 x^4 y z^4 \sqrt{2 y z}$. Now, let's evaluate the given options to identify the equivalent expression. This process involves comparing our simplified result with each of the provided choices, ensuring that all terms inside and outside the radical match. Evaluating options critically is a fundamental skill in mathematics and problem-solving, as it requires a careful examination of each alternative to identify the correct solution. This step reinforces the importance of accuracy and attention to detail in mathematical manipulations.

Option A is $2 x^2 z^2 \sqrt{8 y^3 z}$. This option is incorrect because the terms outside the radical do not match our simplified expression, particularly the coefficients and exponents of $x$ and $z$. Additionally, the expression under the radical is different, as it contains $y^3$ instead of $y$. This comparison highlights the importance of carefully checking each term and exponent when evaluating mathematical expressions. The ability to quickly identify discrepancies is crucial for efficient problem-solving.

Option B is $4 x^2 y z^3 \sqrt{2 x^2}$. This option is also incorrect. The terms outside the radical do not align with our simplified expression, and there is an $x^2$ term under the radical that should not be present. This discrepancy further illustrates the need for meticulous comparison, ensuring that both the coefficients and variables match exactly. Recognizing these differences is a key skill in algebraic manipulation.

Option C is $8 x^4 y z^4 \sqrt{2 y z}$. This option perfectly matches our simplified expression. The terms outside the radical ($8 x^4 y z^4$) and the terms under the radical ($2 y z$) are identical to our result. This match confirms that Option C is the correct equivalent expression. The ability to recognize a perfect match after simplification is a satisfying conclusion to the problem-solving process.

Option D is $64 x^4 y z^4 \sqrt{2 y z}$. This option is incorrect because the coefficient outside the radical is 64, while our simplified expression has a coefficient of 8. Although the variable terms and the expression under the radical match, the difference in the coefficient disqualifies this option. This comparison emphasizes the importance of paying attention to every detail in the expression, including the numerical coefficients. A small difference can lead to an incorrect answer, highlighting the need for precision in mathematical work.

Conclusion: The Equivalent Expression

In conclusion, by systematically simplifying the expression $\sqrt{128 x^8 y^3 z^9}$, we found that the equivalent expression is $8 x^4 y z^4 \sqrt{2 y z}$. This was achieved by breaking down the expression into its prime factors and perfect squares, extracting the square roots where possible, and then comparing the simplified form with the given options. This methodical approach underscores the importance of a step-by-step strategy in solving mathematical problems, allowing for greater accuracy and understanding. The ability to simplify radical expressions is a valuable skill in various mathematical contexts, from algebra to calculus and beyond.

The correct answer is C. $8 x^4 y z^4 \sqrt{2 y z}$. This result highlights the effectiveness of our simplification process and the importance of careful evaluation of options. By understanding the properties of radicals and exponents, and by applying a structured approach, we were able to confidently identify the equivalent expression. This process not only provides the correct answer but also reinforces fundamental mathematical concepts, building a stronger foundation for future problem-solving.

The process of simplifying radicals involves identifying perfect square factors, both numerical and variable, and extracting their square roots. This skill is not only useful in academic settings but also has practical applications in various fields, such as engineering, physics, and computer science. The ability to manipulate and simplify mathematical expressions is a cornerstone of quantitative reasoning and problem-solving, enabling individuals to tackle complex challenges with confidence and precision. Therefore, mastering the techniques of simplifying radicals is a valuable investment in one's mathematical proficiency and analytical capabilities.