Solving For 15pq When P-th Root Of 2 Times Q-th Root Of 5 Equals 4000
Introduction
In this detailed exploration, we will delve into the mathematical problem: if √(p)(2) * √(q)(5) = 4000, then what is the value of 15pq? This problem, which blends the concepts of roots and algebraic manipulation, requires a methodical approach to unravel its intricacies. Our discussion will meticulously break down each step, ensuring a clear understanding of the underlying principles and techniques involved. This article serves as a comprehensive guide, catering not only to those seeking the solution but also to math enthusiasts eager to enhance their problem-solving skills. We will begin by simplifying the given equation and then systematically work towards finding the values of p and q, ultimately leading us to the determination of 15pq. This journey through algebraic territory promises to be both enlightening and rewarding.
Understanding the Problem
Before diving into the solution, let's first dissect the problem statement to ensure we grasp each element fully. The equation we're dealing with is √(p)(2) * √(q)(5) = 4000. Here, √(p)(2) represents the pth root of 2, and √(q)(5) represents the qth root of 5. The goal is to find the value of 15pq. This involves manipulating the given equation to isolate p and q, which can then be used to calculate the final expression. Understanding the properties of roots and exponents is crucial here. The problem subtly tests our ability to convert radical expressions into exponential forms, which often simplifies algebraic manipulations. For instance, the pth root of 2 can be written as 2^(1/p), and similarly, the qth root of 5 can be expressed as 5^(1/q). This conversion is a cornerstone in simplifying the equation and making it more manageable. Furthermore, recognizing that 4000 can be factored into powers of 2 and 5 is another key insight that can guide our solution process. By carefully considering these initial observations, we set the stage for a methodical and effective problem-solving strategy.
Converting to Exponential Form
The initial step in solving the equation √(p)(2) * √(q)(5) = 4000 involves converting the radical expressions into their equivalent exponential forms. This transformation is pivotal because it allows us to apply the rules of exponents, thereby simplifying the equation and making it easier to manipulate. As previously mentioned, the pth root of 2, denoted as √(p)(2), can be rewritten as 2^(1/p). Similarly, the qth root of 5, denoted as √(q)(5), can be expressed as 5^(1/q). By making these substitutions, our equation transforms into 2^(1/p) * 5^(1/q) = 4000. This new form is more amenable to algebraic manipulation. The next step involves expressing 4000 as a product of its prime factors. This is essential because it allows us to equate the powers of the same base on both sides of the equation. Recognizing that 4000 can be written as 2^5 * 5^3 is a crucial observation. With this prime factorization in hand, we can rewrite our equation as 2^(1/p) * 5^(1/q) = 2^5 * 5^3. This form clearly aligns the powers of 2 and 5 on both sides, paving the way for equating exponents and solving for p and q. This conversion to exponential form is not just a symbolic manipulation; it's a strategic move that significantly simplifies the problem, allowing us to leverage the well-established rules of exponents.
Prime Factorization of 4000
To effectively solve the equation 2^(1/p) * 5^(1/q) = 4000, a critical step is to express 4000 as a product of its prime factors. Prime factorization is the process of breaking down a number into a product of its prime number components. This is a fundamental concept in number theory and is invaluable in simplifying equations involving exponents and roots. The prime factorization of 4000 is derived by repeatedly dividing the number by its smallest prime factor until we are left with 1. Starting with 4000, we can first divide by 2, yielding 2000. Dividing 2000 by 2 gives us 1000. Continuing this process, we get 500, 250, and 125. Now, 125 is not divisible by 2, so we move to the next prime number, which is 5. Dividing 125 by 5 gives us 25, and dividing 25 by 5 results in 5. Finally, dividing 5 by 5 leaves us with 1, indicating that we have completed the prime factorization. Counting the number of times each prime factor appears, we find that 2 appears 5 times (2 * 2 * 2 * 2 * 2) and 5 appears 3 times (5 * 5 * 5). Therefore, the prime factorization of 4000 can be expressed as 2^5 * 5^3. This factorization is crucial because it allows us to rewrite the original equation as 2^(1/p) * 5^(1/q) = 2^5 * 5^3. By expressing both sides of the equation in terms of the same prime bases, we can directly compare the exponents, which is the key to solving for p and q. The ability to perform prime factorization is not only useful in this specific problem but is also a fundamental skill in various mathematical contexts, including simplifying fractions, finding greatest common divisors, and solving Diophantine equations.
Equating Exponents
With the equation now in the form 2^(1/p) * 5^(1/q) = 2^5 * 5^3, the next crucial step is to equate the exponents of the same bases on both sides of the equation. This technique is rooted in the fundamental property of exponential functions: if a^m = a^n, then m = n, provided that a is a positive number not equal to 1. This principle allows us to transform a complex exponential equation into simpler algebraic equations. In our case, we have two bases, 2 and 5, on both sides of the equation. By equating the exponents of 2, we get 1/p = 5. Similarly, by equating the exponents of 5, we obtain 1/q = 3. These two equations are much simpler to solve than the original equation. They directly relate the reciprocals of p and q to known values, making it straightforward to find the values of p and q. This step of equating exponents is a powerful technique in solving exponential equations, and it highlights the importance of expressing numbers in their prime factorized form. By recognizing the underlying mathematical principles and applying them systematically, we can efficiently break down complex problems into more manageable parts. The process of equating exponents not only simplifies the equation but also provides a clear pathway to finding the unknown variables, which in this case are p and q. This method is widely used in various areas of mathematics and is an essential tool in any problem-solver's arsenal.
Solving for p and q
Having equated the exponents in the equation 2^(1/p) * 5^(1/q) = 2^5 * 5^3, we now have two simple equations to solve: 1/p = 5 and 1/q = 3. These equations are linear in terms of 1/p and 1/q, making them straightforward to solve for p and q. To solve for p, we take the reciprocal of both sides of the equation 1/p = 5. This gives us p = 1/5. Similarly, to solve for q, we take the reciprocal of both sides of the equation 1/q = 3, which yields q = 1/3. These values of p and q are crucial as they are the key to finding the value of the expression 15pq, which is the ultimate goal of the problem. It is important to note that the values of p and q are fractions, which might seem counterintuitive at first, given that they represent the indices of the roots in the original equation. However, these values are perfectly valid within the context of the problem and the mathematical operations we have performed. The process of solving for p and q demonstrates the power of algebraic manipulation and the importance of understanding the properties of reciprocals. By systematically applying these principles, we can efficiently find the solutions to seemingly complex equations. Now that we have determined the values of p and q, we are well-positioned to calculate the final answer by substituting these values into the expression 15pq.
Calculating 15pq
With the values of p and q determined as p = 1/5 and q = 1/3, the final step in solving the problem is to calculate the value of the expression 15pq. This is a straightforward substitution and multiplication process. We substitute the values of p and q into the expression 15pq, which gives us 15 * (1/5) * (1/3). To simplify this expression, we first multiply 15 by 1/5, which results in 3. So, the expression becomes 3 * (1/3). Next, we multiply 3 by 1/3, which equals 1. Therefore, the value of 15pq is 1. This result provides a conclusive answer to the original problem. The calculation demonstrates the importance of accuracy in each step of the problem-solving process. A small error in any of the previous steps could lead to an incorrect final answer. The simplicity of this final calculation belies the complexity of the steps required to arrive at the correct values of p and q. This underscores the significance of a methodical and systematic approach to problem-solving, where each step builds upon the previous one. By carefully applying algebraic principles and performing accurate calculations, we have successfully navigated through the problem and arrived at the final solution. The value of 15pq being 1 completes our journey, providing a clear and concise answer to the initial question.
Conclusion
In conclusion, we have successfully navigated through the mathematical problem: if √(p)(2) * √(q)(5) = 4000, then what is the value of 15pq? Our journey began with a careful examination of the problem statement, ensuring a clear understanding of the terms and the goal. We then converted the radical expressions into exponential forms, a crucial step that allowed us to apply the rules of exponents. The prime factorization of 4000 played a pivotal role in simplifying the equation, enabling us to equate the exponents of the same bases on both sides. This led us to two simple equations, 1/p = 5 and 1/q = 3, which we solved to find p = 1/5 and q = 1/3. Finally, we substituted these values into the expression 15pq, resulting in the answer 1. This problem-solving process highlights the importance of a systematic approach, where each step is logically connected and builds upon the previous one. It also underscores the significance of understanding fundamental mathematical concepts, such as the properties of roots, exponents, and prime factorization. By breaking down a complex problem into smaller, more manageable steps, we can efficiently arrive at the solution. This exercise not only provides the answer to a specific question but also enhances our problem-solving skills, making us better equipped to tackle similar challenges in the future. The solution, 15pq = 1, is a testament to the power of algebraic manipulation and the beauty of mathematical reasoning.