Solving (3^(x+1) + 2 * 3^(x+2)) / (7 * 3^x) A Step-by-Step Guide
Introduction to Exponential Equations
In the realm of mathematics, exponential equations hold a significant position, particularly in areas like calculus, algebra, and number theory. These equations involve variables in the exponents, demanding a nuanced understanding of mathematical principles for their resolution. This article delves into the intricate solution of the equation (3^(x+1) + 2 * 3^(x+2)) / (7 * 3^x), providing a step-by-step breakdown to enhance comprehension. The core of solving exponential equations lies in the adept application of exponent rules and algebraic manipulations. These rules, which govern how exponents interact under various operations, are the bedrock of simplifying and solving these equations. For instance, the product of powers rule, which states that a^(m) * a^(n) = a^(m+n), and the quotient of powers rule, stating that a^(m) / a^(n) = a^(m-n), are frequently employed. Similarly, the power of a power rule, (a(m))(n) = a^(mn), and the power of a product rule, (ab)^(n) = a(n)b(n), play crucial roles in simplifying complex expressions. In the equation at hand, we will leverage these rules to decompose the exponential terms, thereby paving the way for simplification. Specifically, we will focus on breaking down terms like 3^(x+1) and 3^(x+2) into forms that allow for factoring and cancellation. Furthermore, the algebraic manipulation of exponential equations often involves factoring out common terms. This technique is instrumental in reducing the complexity of the equation and revealing underlying structures that facilitate solving for the variable. By identifying and factoring out common exponential expressions, we can transform the equation into a more manageable form. This process not only simplifies the equation but also helps in isolating the variable, which is the ultimate goal of solving any equation. Throughout this exploration, we will emphasize the importance of precision in applying mathematical operations and the strategic use of algebraic techniques. The combination of these skills is essential for navigating the complexities of exponential equations and arriving at accurate solutions. This article aims to equip readers with a robust understanding of how to approach and solve such equations, enhancing their mathematical proficiency and problem-solving capabilities. The step-by-step approach will illuminate the underlying logic and methodology, making the process accessible and clear. By mastering these techniques, readers can confidently tackle a wide array of exponential equations, solidifying their foundation in this crucial area of mathematics.
Detailed Solution Steps
Step 1: Decompose Exponential Terms
Our initial equation is (3^(x+1) + 2 * 3^(x+2)) / (7 * 3^x). The first strategic move is to decompose the exponential terms in the numerator. This involves expressing 3^(x+1) and 3^(x+2) in terms of 3^x, which will reveal common factors and simplify the equation. Applying the exponent rule a^(m+n) = a^(m) * a^(n), we can rewrite 3^(x+1) as 3^x * 3^1 and 3^(x+2) as 3^x * 3^2. This decomposition is crucial because it isolates the term 3^x, which appears in both terms of the numerator and the denominator. By identifying this common element, we set the stage for factoring, a key technique in simplifying algebraic expressions. The decomposition not only simplifies the equation visually but also highlights the underlying structure, making it easier to see how the terms interact. This step is a testament to the power of exponent rules in transforming complex expressions into more manageable forms. It is a fundamental technique that is applicable across a wide range of exponential equations. By mastering this step, one can effectively reduce the complexity of equations and pave the way for further simplification. The ability to decompose exponential terms is not merely a mechanical skill; it demonstrates a deeper understanding of how exponents work and how they can be manipulated to solve problems. It requires a keen eye for recognizing patterns and applying the appropriate rules. This understanding is the cornerstone of solving exponential equations and is essential for building a strong foundation in algebra and calculus. As we proceed with the solution, the significance of this initial decomposition will become even clearer, as it forms the basis for subsequent simplifications and the ultimate solution of the equation. By breaking down the exponential terms, we create an opportunity to factor out common elements, thereby reducing the equation to a more tractable form. This is a prime example of how strategic application of mathematical principles can lead to elegant solutions.
Step 2: Factor out the Common Term
After decomposing the exponential terms, our equation now looks like (3^x * 3^1 + 2 * 3^x * 3^2) / (7 * 3^x). The next critical step is to factor out the common term, 3^x, from the numerator. Factoring is a powerful algebraic technique that simplifies expressions by extracting common factors. In this case, identifying 3^x as a common factor in both terms of the numerator allows us to rewrite the numerator as 3^x * (3^1 + 2 * 3^2). This transformation is significant because it isolates 3^x as a multiplicative factor, which can then be addressed in the context of the entire equation. The act of factoring not only simplifies the expression but also reveals the underlying structure of the equation. It is a process that reduces complexity and makes it easier to see how different parts of the equation interact. This step is a testament to the importance of recognizing patterns and applying algebraic principles strategically. By mastering the technique of factoring, one can tackle a wide range of mathematical problems more effectively. The ability to identify common factors and extract them from expressions is a fundamental skill in algebra and calculus. It is a tool that allows us to simplify complex equations and make them more amenable to solution. Factoring is not just a mechanical process; it requires a deep understanding of how mathematical operations work and how they can be manipulated to achieve a desired outcome. This understanding is the foundation of problem-solving in mathematics. As we move forward in solving this equation, the impact of factoring out the common term will become even more evident. It sets the stage for the next crucial step, which involves canceling out the common factor in the numerator and the denominator. This cancellation further simplifies the equation and brings us closer to the final solution. By strategically factoring the numerator, we have transformed the equation into a form that is much easier to handle. This is a key illustration of how algebraic techniques can be used to simplify complex expressions and make them more tractable.
Step 3: Simplify by Cancelling Terms
With the numerator now factored as 3^x * (3^1 + 2 * 3^2) and the denominator as 7 * 3^x, the equation stands as (3^x * (3^1 + 2 * 3^2)) / (7 * 3^x). The next crucial step is to simplify the equation by canceling the common term, 3^x, from both the numerator and the denominator. This cancellation is a direct application of the principle that dividing both the numerator and the denominator of a fraction by the same non-zero value does not change the value of the fraction. It's a fundamental simplification technique that drastically reduces the complexity of the equation. Canceling 3^x is particularly effective here because it eliminates the exponential term that was initially causing the most complexity. This step transforms the equation from one involving exponential terms to a simple arithmetic expression. This simplification is not merely a mechanical operation; it reveals the underlying mathematical structure of the equation. By removing the common factor, we gain a clearer view of the remaining elements and their relationships. This is a key principle in mathematical problem-solving: simplification often leads to clarity. The act of canceling common terms is a cornerstone of algebraic manipulation and is applicable across a wide range of mathematical problems. It is a skill that every student of mathematics must master. The ability to recognize and cancel common factors is essential for simplifying fractions, solving equations, and performing other algebraic operations. This skill is not just about following rules; it's about understanding the underlying principles of mathematical equivalence. As we move forward, the impact of this simplification will be clear. The equation is now reduced to a simple arithmetic expression, which can be easily evaluated. This is a testament to the power of strategic simplification in mathematical problem-solving. By canceling the common term, we have transformed the equation from a complex exponential expression into a straightforward calculation. This demonstrates how algebraic techniques can be used to break down complex problems into simpler, more manageable steps.
Step 4: Evaluate the Arithmetic Expression
After canceling the common term, 3^x, our equation has been significantly simplified to (3^1 + 2 * 3^2) / 7. The final step in solving this equation is to evaluate the remaining arithmetic expression. This involves performing the arithmetic operations in the correct order, following the standard mathematical convention of precedence (PEMDAS/BODMAS). First, we calculate the exponential term 3^2, which equals 9. Then, we perform the multiplication 2 * 9, which equals 18. Now, the expression in the numerator becomes 3^1 + 18, which simplifies to 3 + 18, and further simplifies to 21. Finally, we divide this result by the denominator, 7, to get 21 / 7, which equals 3. This sequence of calculations demonstrates the importance of following the correct order of operations. Failing to do so can lead to an incorrect result. The evaluation of arithmetic expressions is a fundamental skill in mathematics and is essential for solving equations, simplifying expressions, and performing calculations in various mathematical contexts. This final step transforms the simplified equation into a single numerical value, which represents the solution to the original problem. It's a culmination of all the previous steps, demonstrating how strategic simplification and algebraic manipulation can lead to a clear and concise answer. The ability to evaluate arithmetic expressions accurately and efficiently is a key skill in mathematics. It is not just about performing calculations; it's about understanding the underlying mathematical principles and applying them correctly. This skill is essential for success in mathematics and other fields that rely on mathematical reasoning. As we have seen, the journey from the original equation to the final solution involves a series of strategic simplifications and calculations. Each step builds upon the previous one, leading to a clear and logical solution. This process highlights the importance of problem-solving skills in mathematics and the power of breaking down complex problems into smaller, more manageable steps.
Final Result and Conclusion
Following the steps of decomposing exponential terms, factoring out the common term, simplifying by canceling, and evaluating the arithmetic expression, we arrive at the final result: 3. This concise numerical answer is the solution to the equation (3^(x+1) + 2 * 3^(x+2)) / (7 * 3^x). The process of solving this equation underscores the importance of mastering fundamental mathematical principles, particularly those related to exponents and algebraic manipulation. The application of exponent rules, such as a^(m+n) = a^(m) * a^(n), allows us to decompose complex exponential terms into simpler forms. This decomposition is a critical first step in simplifying the equation and making it more amenable to solution. Factoring out common terms is another key technique that plays a pivotal role in the simplification process. By identifying and extracting common factors, we can reduce the complexity of expressions and reveal underlying structures. In this case, factoring out 3^x from the numerator allows us to cancel it with the same term in the denominator, leading to a significant simplification. Canceling common terms is a powerful simplification technique that is widely used in algebra and calculus. It is based on the principle that dividing both the numerator and the denominator of a fraction by the same non-zero value does not change the value of the fraction. This technique allows us to eliminate unnecessary complexity and focus on the essential elements of the equation. Evaluating the arithmetic expression is the final step in solving the equation. This involves performing the arithmetic operations in the correct order, following the standard mathematical convention of precedence. By carefully performing these operations, we arrive at the final numerical answer. In conclusion, solving the equation (3^(x+1) + 2 * 3^(x+2)) / (7 * 3^x) is a testament to the power of strategic simplification and algebraic manipulation. By applying fundamental mathematical principles, we can transform complex equations into simpler forms and arrive at clear and concise solutions. This process highlights the importance of mastering these principles and developing strong problem-solving skills in mathematics. The solution not only provides a numerical answer but also offers insights into the underlying mathematical concepts and techniques. It demonstrates how a systematic approach, combined with a solid understanding of mathematical principles, can lead to success in solving complex problems. This understanding is invaluable for students and practitioners of mathematics alike.
