Evaluating (1^3 + 2^3 + 3^3 + 4^3)^(-3/2) A Step-by-Step Guide
Introduction: Unraveling the Power of Sums and Exponents
In the realm of mathematics, we often encounter expressions that seem complex at first glance. However, by carefully dissecting each component and applying fundamental principles, we can unveil the underlying simplicity and arrive at elegant solutions. This article delves into the mathematical expression (1³ + 2³ + 3³ + 4³)^(-3/2), providing a comprehensive explanation of the steps involved in its evaluation. We will explore the concepts of sums of cubes, exponents, and negative powers, ensuring a clear understanding for readers of all backgrounds. Our journey will not only focus on the calculation but also on the reasoning behind each step, highlighting the beauty and logic inherent in mathematical problem-solving.
The core of our exploration lies in understanding the order of operations and how different mathematical concepts intertwine. We begin by addressing the sum of cubes within the parentheses, a foundational arithmetic operation. Subsequently, we navigate the intricacies of exponents, both positive and negative, revealing how they transform numerical values. The final step involves simplifying the expression to its most concise form, demonstrating the power of mathematical manipulation. Throughout this discussion, we aim to foster an appreciation for the elegance and precision of mathematics, encouraging readers to embrace the challenges and rewards it offers. Let us embark on this mathematical adventure, demystifying the expression and unlocking its inherent value.
Breaking Down the Expression: A Step-by-Step Approach
To effectively tackle the expression (1³ + 2³ + 3³ + 4³)^(-3/2), we must adopt a systematic approach, breaking it down into manageable parts. Our initial focus is on the sum of cubes within the parentheses: 1³ + 2³ + 3³ + 4³. This involves calculating the cube of each number individually and then summing the results. Recall that the cube of a number is simply the number multiplied by itself three times. Therefore, 1³ equals 1 * 1 * 1, 2³ equals 2 * 2 * 2, and so on. This seemingly straightforward calculation forms the bedrock of our entire evaluation.
Once we've computed the sum of cubes, we turn our attention to the exponent: -3/2. This exponent introduces two key concepts: a negative sign and a fractional value. The negative sign indicates that we'll be dealing with the reciprocal of the base raised to the corresponding positive exponent. In simpler terms, a^(-n) is equivalent to 1/(a^n). The fractional exponent, in this case, 3/2, represents a combination of a power and a root. The numerator (3) signifies the power to which the base is raised, while the denominator (2) indicates the root to be taken. Specifically, a^(3/2) can be interpreted as the square root of a cubed, or (√a)³. Understanding these nuances of exponents is crucial for accurately simplifying the expression.
By meticulously addressing each component – the sum of cubes and the fractional exponent – we pave the way for a clear and concise solution. This step-by-step methodology not only simplifies the calculation but also enhances comprehension, allowing us to appreciate the underlying mathematical principles at play. The journey through this expression is a testament to the power of methodical problem-solving, a skill that extends far beyond the realm of mathematics.
Calculating the Sum of Cubes: Unveiling the Initial Value
Our first task in evaluating the expression (1³ + 2³ + 3³ + 4³)^(-3/2) is to calculate the sum of the cubes within the parentheses. This involves finding the cube of each number from 1 to 4 and then adding those cubes together. Let's break this down step-by-step: Firstly, 1³ is simply 1 * 1 * 1, which equals 1. Secondly, 2³ is 2 * 2 * 2, resulting in 8. Thirdly, 3³ is 3 * 3 * 3, which gives us 27. Finally, 4³ is 4 * 4 * 4, equaling 64. So, we have the individual cubes: 1, 8, 27, and 64.
Now, we sum these values: 1 + 8 + 27 + 64. Adding these numbers together, we get 1 + 8 = 9, then 9 + 27 = 36, and finally, 36 + 64 = 100. Therefore, the sum of the cubes, 1³ + 2³ + 3³ + 4³, equals 100. This calculation represents a crucial first step in simplifying the overall expression. By accurately determining this initial value, we lay the groundwork for subsequent operations involving exponents and reciprocals. This seemingly simple arithmetic operation underscores the importance of precision in mathematics, where each step builds upon the previous one.
With the sum of cubes now calculated as 100, we can rewrite our original expression as (100)^(-3/2). This transformation allows us to focus on the exponent, which involves both a negative sign and a fractional value. Understanding how to handle these types of exponents is paramount to correctly solving the problem. The next phase of our exploration will delve into the intricacies of negative and fractional exponents, demonstrating their impact on the base value and how they contribute to the final solution. The journey continues, with each step revealing a deeper understanding of the mathematical principles at play.
Delving into Exponents: Unraveling the Power of -3/2
Having determined that the sum of cubes within the parentheses is 100, our expression now stands as (100)^(-3/2). The next critical step involves understanding and applying the exponent -3/2. This exponent embodies two key mathematical concepts: a negative sign and a fractional power. The negative sign indicates that we are dealing with the reciprocal of the base raised to the corresponding positive exponent. In other words, x^(-n) is equivalent to 1/(x^n). Therefore, (100)^(-3/2) can be rewritten as 1 / (100^(3/2)).
The fractional exponent 3/2 signifies a combination of a power and a root. The numerator, 3, represents the power to which the base is raised, while the denominator, 2, indicates the root to be taken. Specifically, x^(3/2) can be interpreted as the square root of x cubed, or (√x)³. Applying this to our expression, 100^(3/2) means we need to find the square root of 100 and then cube the result. The square root of 100 is 10, as 10 * 10 = 100. Cubing 10, we get 10 * 10 * 10, which equals 1000.
Therefore, 100^(3/2) equals 1000. Now, we can substitute this value back into our expression. Recall that (100)^(-3/2) is equivalent to 1 / (100^(3/2)). Since 100^(3/2) is 1000, we have 1 / 1000. This fraction represents the simplified form of our original expression. By carefully dissecting the exponent -3/2 and applying the rules of negative and fractional powers, we've successfully navigated this crucial step in the evaluation. This process highlights the importance of mastering exponent rules, as they form a cornerstone of algebraic manipulation.
Simplifying to the Final Solution: From Expression to Value
Having navigated the complexities of the sum of cubes and the fractional exponent, we've arrived at a crucial juncture: simplifying the expression to its final numerical value. We've determined that (1³ + 2³ + 3³ + 4³)^(-3/2) is equivalent to 1 / 1000. This fraction represents the ultimate simplification of the original expression. To fully grasp the significance of this result, it's essential to understand what this fraction represents in decimal form.
The fraction 1 / 1000 signifies one part out of a thousand. Converting this to decimal notation is straightforward: 1 divided by 1000 equals 0.001. This decimal value represents the final, simplified answer to our mathematical problem. It is the culmination of all the steps we've undertaken, from calculating the sum of cubes to applying the rules of exponents. This journey underscores the elegance of mathematics, where complex expressions can be systematically reduced to concise and meaningful values.
The final answer, 0.001, provides a tangible solution to the initial expression. It showcases the power of mathematical operations and the precision they offer. By carefully applying the order of operations and understanding the properties of exponents, we've successfully demystified a seemingly intricate problem. This process serves as a testament to the importance of a structured approach in mathematics, where each step contributes to the overall solution. The journey from the initial expression to the final value exemplifies the beauty and logic that lie at the heart of mathematics.
Conclusion: The Elegance of Mathematical Simplification
In conclusion, the evaluation of the mathematical expression (1³ + 2³ + 3³ + 4³)^(-3/2) has been a journey through fundamental mathematical concepts, showcasing the elegance of simplification. We began by dissecting the expression into its constituent parts, identifying the sum of cubes and the fractional exponent as key components. The methodical calculation of 1³ + 2³ + 3³ + 4³ led us to the value of 100, setting the stage for the subsequent application of the exponent -3/2.
The exploration of the exponent -3/2 revealed the significance of negative and fractional powers. We understood that the negative sign implied a reciprocal, while the fractional component indicated a combination of a power and a root. By carefully applying these principles, we transformed 100^(-3/2) into 1 / (100^(3/2)), and subsequently, into 1 / 1000. This step-by-step simplification demonstrated the power of exponent rules in maneuvering mathematical expressions.
The final simplification culminated in the fraction 1 / 1000, which we then converted to its decimal equivalent, 0.001. This value represents the ultimate solution to our problem, a concise and meaningful answer derived from a series of logical operations. The journey from the initial expression to this final value highlights the beauty of mathematics, where complex problems can be systematically unravelled and reduced to their simplest forms. This process underscores the importance of a structured approach, a thorough understanding of mathematical principles, and the precision required to arrive at accurate solutions. The expression (1³ + 2³ + 3³ + 4³)^(-3/2), initially appearing complex, ultimately yielded a straightforward answer through the application of mathematical rigor and insight.
