Solving 36^(2x) = 216^(x-1) A Step-by-Step Guide

Introduction

In the realm of mathematics, exponential equations hold a significant place, challenging students and enthusiasts alike to unravel their intricacies. These equations, characterized by variables appearing in the exponents, often require a blend of algebraic manipulation and a solid understanding of exponential properties to solve effectively. This article delves into the process of solving a specific exponential equation: 36^(2x) = 216^(x-1). We will embark on a step-by-step journey, exploring the fundamental principles and techniques involved in finding the solution. Our main focus is to break down each step, ensuring clarity and comprehension, so that you can confidently tackle similar problems in the future. By mastering these techniques, you will not only be able to solve this particular equation but also gain a deeper appreciation for the elegance and power of exponential functions.

Understanding Exponential Equations

Before we dive into the solution, it is important to first understand the basic properties of exponential equations. An exponential equation is one in which the variable appears in the exponent. The key to solving these equations lies in manipulating them so that both sides have the same base. When the bases are the same, we can equate the exponents and solve for the variable. This approach hinges on the fundamental property that if a^m = a^n, then m = n, provided that a is a positive real number not equal to 1. The elegance of this principle allows us to transform complex exponential problems into simpler algebraic equations, which are often easier to manage. Therefore, the primary goal in solving exponential equations is to find a common base, which serves as the bridge connecting both sides of the equation and paving the way for a straightforward solution. This transformation is not always immediately obvious and may require us to factorize the bases or recognize perfect powers. However, once the common base is identified, the subsequent steps become clear and the solution is within reach.

Step 1: Expressing Both Sides with a Common Base

The first crucial step in solving the equation 36^(2x) = 216^(x-1) is to express both sides with a common base. Notice that both 36 and 216 can be expressed as powers of 6. Specifically, 36 = 6^2 and 216 = 6^3. This recognition is essential because it allows us to rewrite the original equation in a more manageable form. By expressing both numbers as powers of 6, we are essentially creating a common language that allows us to compare the exponents directly. The equation now transforms into (6²⁾(2x) = (6³⁾(x-1). This step is not merely a cosmetic change; it is a fundamental move that sets the stage for the rest of the solution. By achieving a common base, we are aligning the mathematical framework to leverage the properties of exponents, making the subsequent steps logically sound and mathematically tractable. This is a powerful technique in solving exponential equations, and mastering this step is key to unlocking more complex problems.

Step 2: Applying the Power of a Power Rule

Now that we have expressed both sides of the equation with a common base, we can apply the power of a power rule. This rule states that (a^m)n = a^(mn)*. Applying this rule to our equation (6²⁾(2x) = (6³⁾(x-1), we get 6^(2 * 2x) = 6^(3 * (x-1)), which simplifies to 6^(4x) = 6^(3x - 3). This step is critical because it simplifies the exponents and brings us closer to equating them. The power of a power rule is a cornerstone in the manipulation of exponential expressions, and its correct application is crucial for solving exponential equations. It effectively streamlines the equation by combining the exponents, making the subsequent steps of equating the exponents and solving for the variable more straightforward. Understanding and correctly applying this rule demonstrates a solid grasp of exponential properties and is an essential skill for anyone tackling exponential equations.

Step 3: Equating the Exponents

With the equation now in the form 6^(4x) = 6^(3x - 3), we can proceed to equate the exponents. This is a direct application of the principle that if a^m = a^n, then m = n. Therefore, we can state that 4x = 3x - 3. This step is the heart of the solution, as it transforms the exponential equation into a linear equation, which is far simpler to solve. By equating the exponents, we are essentially stripping away the exponential framework and focusing on the underlying algebraic relationship between the exponents themselves. This transition is a powerful move, reducing the complexity of the problem to a familiar algebraic form. The ability to make this transition efficiently and correctly is a key indicator of understanding the core principles of exponential equation solving.

Step 4: Solving the Linear Equation

We now have a simple linear equation: 4x = 3x - 3. To solve for x, we need to isolate x on one side of the equation. Subtracting 3x from both sides gives us 4x - 3x = 3x - 3 - 3x, which simplifies to x = -3. This step involves basic algebraic manipulation, a skill essential for solving any mathematical equation. The process of isolating the variable involves applying inverse operations to both sides of the equation, ensuring that the equality is maintained. In this case, subtracting 3x from both sides was the key move, bringing us one step closer to the final solution. The simplicity of this step underscores the power of the previous transformations, which reduced the initial complex exponential equation to a straightforward linear equation. The solution, x = -3, represents the value that, when substituted back into the original equation, will make the equation true.

Step 5: Verifying the Solution

To ensure that our solution is correct, it is essential to verify it by substituting x = -3 back into the original equation: 36^(2x) = 216^(x-1). Substituting x = -3, we get 36^(2 * -3) = 216^(-3 - 1), which simplifies to 36^(-6) = 216^(-4). Now, we can express both sides as powers of 6: (6²⁾(-6) = (6³⁾(-4). Applying the power of a power rule, we have 6^(-12) = 6^(-12). Since both sides are equal, our solution x = -3 is verified. This verification step is crucial in the problem-solving process, as it provides a check against potential errors made during the algebraic manipulations. It reinforces the correctness of the solution and demonstrates a thorough understanding of the problem-solving process. By verifying the solution, we can be confident that our answer is accurate and that we have successfully solved the exponential equation.

Conclusion

In conclusion, solving exponential equations involves a systematic approach that combines understanding exponential properties with algebraic manipulation. In the specific case of 36^(2x) = 216^(x-1), we successfully found the solution x = -3 by expressing both sides with a common base, applying the power of a power rule, equating the exponents, solving the resulting linear equation, and verifying the solution. The process demonstrated the importance of breaking down a complex problem into smaller, manageable steps. Each step, from identifying a common base to verifying the solution, plays a critical role in arriving at the correct answer. The mastery of these techniques not only enables the solution of specific equations but also builds a solid foundation for tackling more complex mathematical problems. The journey through this problem highlights the interconnectedness of mathematical concepts and the power of a structured approach in problem-solving. By practicing these methods and understanding the underlying principles, one can confidently approach and solve a wide range of exponential equations.

Final Answer

The final answer to the equation 36^(2x) = 216^(x-1) is x = -3.