Solving For K In 50100=k(10050) A Step-by-Step Guide

Introduction: Deciphering the Mathematical Puzzle

In the realm of mathematics, equations often present themselves as puzzles, challenging us to unravel the unknown and discover the hidden values. Today, we embark on a journey to solve one such intriguing equation: 50100 = k(10050). Our mission is to determine the value of the elusive variable k, carefully navigating through the mathematical landscape to arrive at the correct solution. This problem delves into the fascinating world of exponents and their properties, requiring a keen understanding of how numbers behave when raised to powers. As we dissect the equation, we'll explore different strategies and techniques to isolate k and unveil its true identity. Prepare yourself to delve deep into the heart of this mathematical exploration, where each step brings us closer to the ultimate answer.

Deconstructing the Equation: A Step-by-Step Approach

To effectively tackle the equation 50100 = k(10050), we must first dissect it into its fundamental components. We begin by recognizing that we are dealing with exponents, where numbers are raised to a certain power. In this case, we have 50 raised to the power of 100 and 100 raised to the power of 50. The variable k acts as a scaling factor, connecting these two exponential expressions. Our primary objective is to isolate k on one side of the equation, allowing us to express it in terms of known quantities. This involves strategic manipulation of the equation, employing the rules of exponents and algebraic principles. We'll carefully analyze the relationship between the bases (50 and 100) and the exponents (100 and 50), seeking opportunities to simplify the expression and reveal the value of k. This methodical approach will ensure that we navigate the complexities of the equation with precision and clarity.

Expressing 50100 in Terms of its Prime Factors

The first crucial step in solving this equation is to express 50100 in terms of its prime factors. This allows us to break down the number into its most fundamental components, making it easier to manipulate and compare with other exponential expressions. We know that 50 can be factored as 2 * 5 * 5, or 2 * 52. Therefore, 50100 can be written as (2 * 52)100. Now, we apply the power of a product rule, which states that (ab)n = anbn. This gives us 2100 * (52)100. Further simplifying, we use the power of a power rule, (am)n = amn, to get 2100 * 5200. This prime factorization of 50100 provides us with a clearer understanding of its structure and sets the stage for further simplification.

Expressing 10050 in Terms of its Prime Factors

Similarly, we need to express 10050 in terms of its prime factors. This will allow us to establish a common ground with the expression for 50100 and facilitate the isolation of the variable k. We know that 100 can be factored as 2 * 2 * 5 * 5, or 22 * 52. Therefore, 10050 can be written as (22 * 52)50. Applying the power of a product rule, we get (22)50 * (52)50. Using the power of a power rule, we further simplify this to 2100 * 5100. Now, we have a clear picture of the prime factorization of 10050, which is essential for comparing it with the prime factorization of 50100.

Isolating k and Determining its Value

Now that we have expressed both 50100 and 10050 in terms of their prime factors, we can proceed to isolate k and determine its value. Recall the original equation: 50100 = k(10050). We have established that 50100 = 2100 * 5200 and 10050 = 2100 * 5100. Substituting these expressions into the original equation, we get: 2100 * 5200 = k(2100 * 5100). To isolate k, we divide both sides of the equation by (2100 * 5100): k = (2100 * 5200) / (2100 * 5100). Now, we can simplify this expression by canceling out the common factor of 2100: k = 5200 / 5100. Applying the quotient of powers rule, which states that am / an = am-n, we get: k = 5200-100 = 5100. Therefore, the value of k is 5100, which corresponds to option (b).

Analyzing the Solution: 5100

Having determined that k = 5100, it's crucial to analyze this solution in the context of the original equation. We found this value by meticulously breaking down the exponential expressions, utilizing prime factorization and the rules of exponents. The result, 5100, represents a significant number, highlighting the power of exponents to generate large values. This solution confirms the relationship between 50100 and 10050, demonstrating how the scaling factor k bridges the gap between these two exponential quantities. Moreover, this exercise underscores the importance of understanding the fundamental properties of exponents and prime factorization in solving mathematical problems. The process of arriving at this solution has not only provided us with the answer but also deepened our appreciation for the elegance and precision of mathematical reasoning.

Exploring Alternative Approaches

While we successfully determined the value of k using prime factorization and the rules of exponents, it's beneficial to consider alternative approaches. Exploring different methods can provide a more comprehensive understanding of the problem and potentially offer alternative pathways to the solution. One such approach involves directly manipulating the exponents using logarithmic properties. By taking the logarithm of both sides of the equation, we can transform the exponential equation into a linear one, making it easier to isolate k. Another approach involves recognizing the relationship between 50 and 100, noting that 100 is simply twice 50. This observation can lead to a more direct simplification of the equation, potentially bypassing the need for extensive prime factorization. By considering these alternative approaches, we not only reinforce our problem-solving skills but also gain a broader perspective on the mathematical landscape.

Conclusion: The Significance of Exponents

In conclusion, the equation 50100 = k(10050) has served as a compelling example of the power and intricacies of exponents in mathematics. Through a step-by-step process, we successfully determined that k = 5100. This journey involved expressing numbers in terms of their prime factors, applying the rules of exponents, and strategically manipulating the equation to isolate the unknown variable. The solution highlights the fundamental relationship between exponential expressions and the importance of understanding their properties. Moreover, this exercise has underscored the value of methodical problem-solving, emphasizing the need to break down complex equations into manageable steps. As we conclude this exploration, we carry with us a deeper appreciation for the elegance and power of exponents in the world of mathematics.