Simplify (x^(1/2)y^(-1/4)z)^(-2) A Step-by-Step Guide

In the realm of mathematics, simplifying complex expressions is a fundamental skill. This article delves into the intricacies of simplifying the expression (x^(1/2)y(-1/4)z)^(-2), providing a step-by-step guide to arrive at the equivalent expression. We will explore the core concepts of exponents and their properties, ultimately leading to the correct simplification. This exploration is not just about finding the answer; it's about understanding the underlying mathematical principles that govern these operations. By grasping these principles, you'll be better equipped to tackle a wide array of algebraic problems. The journey of simplifying this expression is a testament to the power of mathematical rules and their ability to transform seemingly complex forms into more manageable ones.

Understanding the Fundamentals of Exponents

Before we dive into the specific expression, let's solidify our understanding of exponents. An exponent indicates the number of times a base is multiplied by itself. For instance, in the expression x^n, 'x' is the base and 'n' is the exponent. This means 'x' is multiplied by itself 'n' times. Fractional exponents, like the ones we encounter in our expression, represent roots. For example, x^(1/2) signifies the square root of x, and x^(1/4) represents the fourth root of x. Negative exponents, on the other hand, indicate reciprocals. That is, x^(-n) is equivalent to 1/x^n. Mastering these fundamental concepts is crucial for simplifying expressions involving exponents. The ability to seamlessly transition between exponential and radical forms is a key skill in algebraic manipulations. Furthermore, understanding the behavior of negative exponents is essential for correctly handling expressions where variables are raised to negative powers. With a firm grasp of these basics, we can confidently approach the simplification of more complex expressions.

Key Properties of Exponents

To effectively simplify expressions with exponents, it's essential to be familiar with the key properties that govern their behavior. These properties provide the rules for manipulating exponents in various situations. One of the most fundamental properties is the power of a power rule, which states that (x^m)n = x^(m*n). This rule is crucial when dealing with expressions where a power is raised to another power, as it allows us to multiply the exponents. Another important property is the product of powers rule, which states that x^m * x^n = x^(m+n). This rule applies when multiplying terms with the same base, where we add the exponents. The quotient of powers rule, x^m / x^n = x^(m-n), is used when dividing terms with the same base, where we subtract the exponents. Lastly, the property of a product to a power, (xy)^n = x^n * y^n, and a quotient to a power, (x/y)^n = x^n / y^n, are vital when dealing with expressions involving products or quotients raised to a power. These properties, when applied correctly, can significantly simplify complex expressions and lead to a more manageable form. Understanding and memorizing these rules is an investment that pays off in countless algebraic manipulations.

Step-by-Step Simplification of (x^(1/2)y(-1/4)z)^(-2)

Now, let's apply these principles to simplify the expression (x^(1/2)y(-1/4)z)^(-2). Our goal is to eliminate the outer exponent and express the result in a simplified form. The first step involves applying the power of a product rule, which dictates that (abc)^n = a^n * b^n * c^n. In our case, this means distributing the exponent -2 to each term inside the parentheses: x^(1/2)(-2) * y^(-1/4)(-2) * z^(-2). This distribution is a critical step, as it allows us to deal with each term individually. Next, we apply the power of a power rule, (x^m)n = x^(m*n), to each term. For x^(1/2)(-2), we multiply the exponents (1/2) and (-2), resulting in x^(-1). Similarly, for y^(-1/4)(-2), we multiply (-1/4) and (-2), which gives us y^(1/2). Finally, z^(-2) remains as it is. So, our expression now looks like: x^(-1) * y^(1/2) * z^(-2). This intermediate step is crucial for breaking down the complex expression into simpler components. By applying the power rules systematically, we have transformed the original expression into a form that is easier to manipulate.

Dealing with Negative Exponents

The next step in simplifying our expression, x^(-1) * y^(1/2) * z^(-2), involves addressing the negative exponents. Recall that a negative exponent indicates a reciprocal. That is, x^(-n) is equivalent to 1/x^n. Applying this rule to our expression, we can rewrite x^(-1) as 1/x and z^(-2) as 1/z^2. The term y^(1/2) remains as it is, since it has a positive exponent. Now, our expression looks like: (1/x) * y^(1/2) * (1/z^2). To combine these terms into a single fraction, we multiply the numerators and the denominators. This gives us y^(1/2) / (x * z^2). This step is a crucial transformation, as it eliminates the negative exponents and presents the expression in a more conventional form. By understanding the relationship between negative exponents and reciprocals, we can effectively manipulate expressions and simplify them to their most basic forms. This manipulation not only simplifies the expression but also makes it easier to interpret and use in further calculations.

The Final Simplified Expression

After dealing with the negative exponents, we have arrived at the expression y^(1/2) / (x * z^2). This is the simplified form of the original expression (x^(1/2)y(-1/4)z)^(-2). We have successfully applied the properties of exponents to eliminate the outer exponent and express the result in a more concise and understandable form. The final expression clearly shows the relationship between the variables and their powers. The variable 'y' is raised to the power of 1/2, indicating its square root, while 'x' and 'z' are in the denominator, with 'z' squared. This simplified form is not only mathematically equivalent to the original expression but also provides a clearer picture of the underlying relationships. It is important to note that this simplified form is often preferred in mathematical contexts, as it is easier to work with and interpret. The process of simplifying expressions is a fundamental skill in algebra and is crucial for solving equations, evaluating functions, and understanding mathematical relationships.

Identifying the Correct Equivalent Expression

Now that we have simplified the expression (x^(1/2)y(-1/4)z)^(-2) to y^(1/2) / (x * z^2), we can confidently identify the correct equivalent expression from the given options. Comparing our simplified expression with the options provided, we can see that option C, y^(1/2) / (x z^2), matches our result perfectly. Options A and B, x^(1/2) / (y z^2) and x^(1/2) / (y^(1/4) z^2), respectively, do not match our simplified expression. This highlights the importance of careful and accurate simplification. By following the correct steps and applying the properties of exponents correctly, we have arrived at the correct answer and eliminated the incorrect options. This process of elimination is a valuable strategy in mathematics, as it allows us to narrow down the possibilities and focus on the most likely solution. Furthermore, the ability to verify our answer by comparing it with the given options provides a crucial check for accuracy and reinforces our understanding of the concepts involved.

Conclusion: Mastering Exponent Simplification

In conclusion, simplifying expressions with exponents is a crucial skill in mathematics. By understanding the fundamental properties of exponents and applying them systematically, we can transform complex expressions into simpler, more manageable forms. In this article, we have successfully simplified the expression (x^(1/2)y(-1/4)z)^(-2) to y^(1/2) / (x * z^2), demonstrating the power of these principles. This process involved distributing the outer exponent, applying the power of a power rule, dealing with negative exponents, and combining terms into a single fraction. The ability to simplify expressions is not only essential for solving mathematical problems but also for developing a deeper understanding of mathematical relationships. By mastering exponent simplification, you will be better equipped to tackle a wide range of algebraic challenges and excel in your mathematical endeavors. This skill is a cornerstone of mathematical proficiency and will serve you well in various fields that rely on mathematical principles.