Simplifying Expressions Equivalent Expression Of (16x^8y^{-12})^(1/2)

Introduction

In this comprehensive exploration, we will embark on a journey to decipher the equivalent expression for the algebraic expression $\left(16 x^8 y^{-12}\right)^{\frac{1}{2}}$. This problem delves into the fundamental principles of exponents and their application in simplifying complex expressions. Our mission is to navigate through the intricacies of exponent rules, ensuring a thorough understanding of each step involved. We will meticulously dissect the given expression, applying relevant properties of exponents to arrive at the simplified form. This process will not only reveal the correct answer but also fortify our grasp of algebraic manipulations. The key to solving this problem lies in the judicious use of the power of a product rule and the power of a power rule, coupled with a keen awareness of how negative exponents behave. By the end of this detailed analysis, you will be well-equipped to tackle similar challenges with confidence and precision. Let's begin our expedition into the realm of exponents and uncover the simplified form of the given expression.

Understanding the Problem

The core of our mathematical quest lies in simplifying the expression $\left(16 x^8 y^{-12}\right)^{\frac{1}{2}}$. This expression presents a combination of numerical coefficients, variables raised to exponents, and a fractional exponent acting upon the entire term. To unravel this, we must invoke the fundamental principles of exponents. Specifically, we'll be leveraging the power of a product rule, which states that $(ab)^n = a^n b^n$, and the power of a power rule, which dictates that $(a^m)^n = a^{mn}$. Furthermore, we need to understand how to handle negative exponents, recalling that $a^{-n} = \frac{1}{a^n}$. The fractional exponent of $\frac{1}{2}$ signifies taking the square root. Therefore, our task involves not only simplifying the exponents but also finding the square root of the numerical coefficient. This requires a methodical approach, carefully applying each rule in the correct sequence. We must pay close attention to the order of operations and ensure that each simplification step is logically sound. By doing so, we can systematically transform the given expression into its most simplified equivalent form, revealing the correct answer among the provided options. The challenge is not merely about arriving at the solution but also about understanding the underlying mathematical principles that govern the transformation.

Step-by-Step Solution

To solve the expression $\left(16 x^8 y^{-12}\right)^{\frac{1}{2}}$, we will meticulously apply the rules of exponents. First, we distribute the exponent $\frac{1}{2}$ to each factor within the parentheses. This utilizes the power of a product rule, which states that $(ab)^n = a^n b^n$. Applying this rule, we get:

$16^{\frac{1}{2}} \cdot (x^8)^{\frac{1}{2}} \cdot (y^{-12})^{\frac{1}{2}}$

Next, we simplify each term individually. The term $16^{\frac{1}{2}}$ represents the square root of 16, which is 4. For the variable terms, we apply the power of a power rule, $(a^m)^n = a^{mn}$. Thus, $(x^8)^{\frac{1}{2}} = x^{8 \cdot \frac{1}{2}} = x^4$, and $(y^{-12})^{\frac{1}{2}} = y^{-12 \cdot \frac{1}{2}} = y^{-6}$. Substituting these simplified terms back into the expression, we have:

$4 \cdot x^4 \cdot y^{-6}$

Now, we address the negative exponent. Recall that $a^{-n} = \frac{1}{a^n}$. Therefore, $y^{-6} = \frac{1}{y^6}$. Substituting this into our expression, we get:

$4 \cdot x^4 \cdot \frac{1}{y^6}$

Finally, we combine the terms to arrive at the simplified expression:

$\frac{4x^4}{y^6}$

This step-by-step approach, utilizing the power of a product rule, the power of a power rule, and the handling of negative exponents, leads us to the equivalent expression. By carefully applying each rule, we transform the original complex expression into a simplified and easily understandable form.

Analyzing the Options

Having derived the simplified expression $\frac{4 x^4}{y^6}$, we now turn our attention to the provided options to identify the correct match. The options presented are:

A. $-4 x^4 y^6$ B. $-8 x^4 y^6$ C. $\frac{4 x^4}{y^6}$ D. $\frac{8 x^4}{y^6}$

By comparing our derived expression with the given options, we can clearly see that option C, $\frac{4 x^4}{y^6}$, precisely matches our simplified form. Options A and B introduce a negative sign and an incorrect power of $y$, while option D has an incorrect coefficient. Therefore, only option C aligns with our calculated result. This process of comparing our solution with the provided choices is a crucial step in problem-solving, ensuring that we select the correct answer. It also reinforces our understanding of the simplification process, as we can see how the other options deviate from the correct solution. By carefully analyzing each option, we solidify our confidence in the accuracy of our answer. This methodical approach not only helps in solving this particular problem but also equips us with a valuable strategy for tackling future mathematical challenges.

Conclusion

In conclusion, through a meticulous application of exponent rules, we have successfully simplified the expression $\left(16 x^8 y^{-12}\right)^{\frac{1}{2}}$ to its equivalent form, $\frac{4 x^4}{y^6}$. This journey involved leveraging the power of a product rule, the power of a power rule, and a thorough understanding of negative exponents. Each step was carefully executed, ensuring a clear and logical progression towards the solution. We began by distributing the fractional exponent, then simplified each term individually, and finally addressed the negative exponent to arrive at the final simplified expression. The comparison with the provided options confirmed that our derived answer, $\frac{4 x^4}{y^6}$, matched option C, solidifying our solution. This exercise underscores the importance of mastering exponent rules and their application in simplifying algebraic expressions. It also highlights the significance of a methodical approach, where each step is carefully considered and executed. By understanding the underlying principles and practicing their application, we can confidently tackle a wide range of mathematical problems. The ability to simplify expressions is a fundamental skill in mathematics, and this problem serves as a valuable illustration of how to effectively apply these principles.

Keywords

Equivalent expression, exponent rules, simplifying expressions, power of a product rule, power of a power rule, negative exponents, fractional exponents, square root, algebraic manipulations