Solving Exponential Equations Finding The Value Of X In 343^x = 49^(4-x)

Introduction

In this article, we will delve into the process of solving exponential equations, specifically focusing on the equation 343^x = 49^(4-x). This type of problem is commonly encountered in algebra and precalculus courses, and mastering the techniques to solve them is crucial for a strong foundation in mathematics. Our primary focus will be on expressing both sides of the equation with the same base, a strategy that simplifies the equation and allows us to solve for the unknown variable, x. This method hinges on the fundamental property that if a^m = a^n, then m = n, provided that a is a positive number not equal to 1. This property allows us to equate the exponents once we have the same base on both sides.

Our goal is to provide a step-by-step guide, ensuring clarity and understanding at each stage. We will begin by identifying a common base for 343 and 49, which in this case is 7. Expressing both numbers as powers of 7 will allow us to rewrite the equation in a simpler form. Next, we will apply the power of a power rule, a key exponent rule that states (a^m)n = a^(mn)*. This rule will help us to further simplify the equation. Once we have the same base on both sides, we can equate the exponents and solve the resulting linear equation for x. Finally, we will verify our solution by substituting it back into the original equation to ensure accuracy. This comprehensive approach will not only lead us to the correct answer but also reinforce the underlying mathematical principles involved.

This article aims to be more than just a solution guide; it is a learning resource. By carefully explaining each step and highlighting the underlying mathematical concepts, we hope to equip readers with the skills and confidence to tackle similar problems independently. Whether you are a student preparing for an exam or simply looking to brush up on your algebra skills, this article will provide a clear and thorough explanation of how to solve exponential equations. Let's embark on this mathematical journey and unravel the value of x in the given equation.

Expressing Both Sides with a Common Base

To effectively tackle the equation 343^x = 49^(4-x), the initial crucial step involves expressing both sides of the equation using a common base. Recognizing the numerical relationships is key here. Both 343 and 49 can be expressed as powers of 7. Specifically, 343 = 7^3 and 49 = 7^2. This recognition is not arbitrary; it stems from understanding the prime factorization of these numbers. By identifying 7 as the common base, we pave the way for simplifying the equation significantly.

Substituting these expressions back into the original equation, we get (7³⁾x = (7²⁾(4-x). This transformation is a pivotal step because it allows us to leverage the properties of exponents in the subsequent stages. The equation now clearly showcases the common base, setting the stage for the application of the power of a power rule. This step underscores the importance of recognizing numerical relationships and expressing numbers in their exponential forms to facilitate simplification in mathematical problem-solving. Furthermore, this strategy is not limited to this specific problem; it's a widely applicable technique in solving various exponential equations. The ability to identify common bases and rewrite equations accordingly is a fundamental skill in algebra.

This initial transformation is not merely about changing the appearance of the equation; it’s about altering its structure to reveal the underlying mathematical relationships more clearly. By expressing both sides with the same base, we are essentially creating a scenario where the exponents can be directly compared, leading us closer to the solution. This step exemplifies the power of mathematical manipulation in simplifying complex problems. The choice of 7 as the base is strategic, driven by the desire to create a more manageable equation. Without this initial step, solving the equation would be significantly more challenging. The common base approach is a cornerstone technique in handling exponential equations, and mastering it is essential for success in algebra and beyond. In the following sections, we will see how this initial step sets the stage for further simplification and ultimately leads to the solution of the equation.

Applying the Power of a Power Rule

Having successfully expressed both sides of the equation 343^x = 49^(4-x) with a common base of 7, transforming it into (7³⁾x = (7²⁾(4-x), the next vital step involves applying the power of a power rule. This rule, a cornerstone of exponent manipulation, states that (a^m)n = a^(mn)*. It's the key to further simplifying our equation and bringing us closer to isolating the variable x. By correctly applying this rule, we can eliminate the parentheses and consolidate the exponents, making the equation much more manageable.

Applying the power of a power rule to the left side of the equation, (7³⁾x becomes 7^(3x). Similarly, on the right side, (7²⁾(4-x) transforms into 7^(2(4-x))*, which simplifies to 7^(8-2x). The equation now reads 7^(3x) = 7^(8-2x). This transformation is a significant leap forward in our problem-solving process. By applying the power of a power rule, we have reduced the complexity of the equation, making it easier to handle. The exponents are now clearly expressed, setting the stage for the next step: equating the exponents.

The power of a power rule is not just a mechanical operation; it reflects a fundamental property of exponents. Understanding why this rule works is just as important as knowing how to apply it. The rule essentially states that raising a power to another power involves multiplying the exponents. This stems from the basic definition of exponents as repeated multiplication. Applying this rule correctly is crucial, as any error in this step can lead to an incorrect solution. The ability to confidently apply the power of a power rule is a hallmark of proficiency in algebra. It's a tool that is used extensively in various mathematical contexts, from solving equations to simplifying expressions.

In this context, the power of a power rule acts as a bridge, connecting the initial expression of the equation to a form where we can directly compare the exponents. It's a testament to the power of mathematical rules in simplifying complex problems. The transformation we have achieved in this step is a critical one, paving the way for the final stages of solving for x. Without the application of the power of a power rule, the equation would remain in a form that is much harder to solve. In the subsequent sections, we will see how equating the exponents and solving the resulting linear equation will lead us to the solution.

Equating Exponents and Solving for x

Having simplified the equation to 7^(3x) = 7^(8-2x), we now arrive at a crucial juncture where we can equate the exponents. This step is justified by 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. In our case, the base is 7, which satisfies this condition. Equating the exponents allows us to transform the exponential equation into a linear equation, which is much easier to solve.

By equating the exponents, we obtain the linear equation 3x = 8 - 2x. This equation is a straightforward algebraic expression that can be solved using basic algebraic techniques. Our goal is to isolate x on one side of the equation. To do this, we can begin by adding 2x to both sides of the equation, which gives us 5x = 8. Now, to isolate x completely, we divide both sides of the equation by 5, resulting in x = 8/5. This is the solution to our equation.

The process of equating exponents is not just a mechanical step; it's a logical deduction based on the properties of exponential functions. Understanding this underlying principle is crucial for mastering the technique. The ability to transform an exponential equation into a linear equation is a powerful tool in algebra. It allows us to solve problems that would otherwise be much more challenging. The resulting linear equation, 3x = 8 - 2x, is a classic example of a linear equation in one variable. Solving such equations is a fundamental skill in algebra, and the techniques involved are widely applicable in various mathematical contexts.

The solution we have obtained, x = 8/5, is a numerical value that satisfies the original equation. However, it's always a good practice to verify our solution by substituting it back into the original equation to ensure accuracy. This step helps to catch any potential errors that may have occurred during the problem-solving process. In the next section, we will perform this verification to confirm that our solution is indeed correct. The process of equating exponents and solving the resulting equation is a key technique in solving exponential equations, and it's a skill that is essential for success in algebra and beyond.

Verifying the Solution

To ensure the accuracy of our solution, x = 8/5, it is essential to verify it by substituting it back into the original equation, 343^x = 49^(4-x). This process not only confirms that our solution is correct but also reinforces our understanding of the equation and the steps we took to solve it. Verification is a crucial step in problem-solving, as it helps to identify any potential errors that may have been made along the way.

Substituting x = 8/5 into the left side of the equation, we get 343^(8/5). Since 343 = 7^3, this can be rewritten as (7³⁾(8/5), which, using the power of a power rule, simplifies to 7^(24/5). Now, let's substitute x = 8/5 into the right side of the equation. We get 49^(4 - 8/5). Since 49 = 7^2, this becomes (7²⁾(4 - 8/5). Simplifying the exponent, 4 - 8/5 = 20/5 - 8/5 = 12/5. So, the right side of the equation becomes (7²⁾(12/5), which, using the power of a power rule, simplifies to 7^(24/5).

Comparing the simplified expressions for both sides of the equation, we see that 7^(24/5) = 7^(24/5). This equality confirms that our solution, x = 8/5, is indeed correct. The verification process demonstrates the consistency of our solution with the original equation. It also provides a sense of closure, assuring us that we have successfully solved the problem. The act of substituting the solution back into the original equation is a valuable learning experience in itself. It reinforces the connections between the different steps in the problem-solving process and deepens our understanding of the underlying mathematical concepts.

In addition to confirming the correctness of our solution, the verification process also highlights the importance of accuracy in each step of the problem-solving process. Any error in the intermediate steps could lead to an incorrect solution, which would be revealed during verification. This underscores the need for careful and methodical work in mathematics. The verification step is not just a formality; it's an integral part of the problem-solving process. It's a way of checking our work and ensuring that we have arrived at the correct answer. In conclusion, the successful verification of our solution reinforces our confidence in the solution and our problem-solving skills.

Conclusion

In this comprehensive exploration, we have successfully navigated the intricacies of solving the exponential equation 343^x = 49^(4-x). By methodically applying the principles of exponents and algebraic manipulation, we have determined that the value of x is 8/5. Our journey began with the crucial step of expressing both sides of the equation using a common base, which in this case was 7. This transformation laid the foundation for simplifying the equation and leveraging the properties of exponents.

We then skillfully applied the power of a power rule, a fundamental concept in exponent manipulation, to further streamline the equation. This rule allowed us to eliminate parentheses and consolidate the exponents, bringing us closer to isolating the variable x. Following this, we employed the key property that if a^m = a^n, then m = n, to equate the exponents. This step transformed the exponential equation into a linear equation, which we readily solved using basic algebraic techniques.

Finally, to ensure the accuracy of our solution, we performed a thorough verification by substituting x = 8/5 back into the original equation. This process not only confirmed the correctness of our solution but also reinforced our understanding of the equation and the steps we took to solve it. The successful verification provided a sense of closure and confidence in our problem-solving abilities.

This exercise highlights the importance of a systematic approach to solving mathematical problems. By breaking down the problem into manageable steps and applying the appropriate mathematical principles, we were able to arrive at the correct solution. The techniques and concepts discussed in this article are widely applicable in various mathematical contexts, making this a valuable learning experience. Mastering the art of solving exponential equations is a crucial step in developing a strong foundation in algebra and precalculus. The skills acquired through this process will serve as a valuable asset in tackling more complex mathematical challenges in the future. We hope this detailed explanation has provided clarity and insight into the process of solving exponential equations and has empowered you to confidently approach similar problems.