Error Analysis And Correction Solving Exponential Equations

Introduction

This article delves into the step-by-step analysis of a mathematical problem involving exponential equations. Our primary focus is to identify the error within the provided solution and to meticulously demonstrate the correct methodology to arrive at the accurate answer. We aim to provide a comprehensive understanding of the underlying principles of exponents and how they apply to solving equations. This detailed explanation will not only pinpoint the mistake but also reinforce the correct techniques, ensuring a clearer grasp of the subject matter.

Problem Statement

The given problem is to solve the exponential equation:

$\frac{1}{64} = 16^{2a}$

The provided solution steps are as follows:

$\begin{array}{l} \frac{1}{64}=16^{2 a} \\ 4^{-3}=\left(2^4\right)^{2 a} \\ 4^{-3}=2^{8 a} \\ -3=8 a \\ -\frac{3}{8}=a \end{array}$

We will dissect each step to find the precise location and nature of the error.

Step-by-Step Error Analysis

Let’s break down the solution step by step:

Step 1: $\frac{1}{64} = 16^{2a}$

This is the original equation, and there is no error here. It simply states the problem we need to solve. This step is the foundation upon which we will build our solution, and it's crucial to start with the correct premise. The equation sets the stage for our exploration of exponential relationships and serves as the starting point for our algebraic manipulations.

Step 2: $4^{-3} = (2^4)^{2a}$

Here, the left side of the equation, $\frac{1}{64}$, is rewritten as $4^{-3}$. This is correct because $4^3 = 64$, and therefore $\frac{1}{64}$ is indeed $4^{-3}$. On the right side, $16$ is rewritten as $2^4$, which is also correct. So, $16^{2a}$ becomes $(2^4)^{2a}$. This transformation is a key step in simplifying the equation, as it expresses both sides in terms of the same base, which is essential for solving exponential equations. By expressing both sides with a common base, we can equate the exponents and solve for the unknown variable. The manipulation in this step is mathematically sound and aligns with the properties of exponents.

Step 3: $4^{-3} = 2^{8a}$

On the right side, $(2^4)^{2a}$ is simplified to $2^{8a}$ using the power of a power rule, which states that $(a^m)^n = a^{mn}$. In this case, $(2^4)^{2a} = 2^{4 imes 2a} = 2^{8a}$. This application of the power of a power rule is accurate and essential for simplifying the equation further. However, the left side remains as $4^{-3}$, which is where the error begins to manifest. To correctly proceed, we should express the left side, $4^{-3}$, in terms of the base 2 as well. Maintaining a consistent base on both sides is crucial for equating the exponents later on. The failure to convert the left side to the same base as the right side is the critical oversight in this step.

Step 4: $-3 = 8a$

This step incorrectly equates the exponents. The equation $4^{-3} = 2^{8a}$ cannot directly lead to $-3 = 8a$ because the bases are different (4 and 2). This is the major error in the solution. Before equating exponents, the bases on both sides of the equation must be the same. This step overlooks the fundamental requirement of having a common base before equating exponents. The exponents can only be directly compared when the bases are identical. This step erroneously jumps to a conclusion without ensuring the necessary condition of equal bases, leading to an incorrect equation.

Step 5: $-\frac{3}{8} = a$

This step solves the incorrect equation $-3 = 8a$ for $a$. While the algebraic manipulation is correct (dividing both sides by 8), the result is based on the erroneous equation from the previous step. Therefore, the value $-\frac{3}{8}$ is not the correct solution to the original problem. This final step, though arithmetically sound, is built upon a flawed foundation, making the resulting value of 'a' incorrect. The solution obtained in this step is a direct consequence of the error made in the earlier steps, highlighting the importance of accurately applying mathematical principles throughout the problem-solving process.

Correct Solution

To correctly solve the equation, we need to express both sides with the same base. The original equation is:

$\frac{1}{64} = 16^{2a}$

Step 1: Express both sides with a base of 2

We know that $64 = 2^6$, so $\frac{1}{64} = 2^{-6}$. Also, $16 = 2^4$, so $16^{2a} = (2^4)^{2a} = 2^{8a}$. Thus, the equation becomes:

$2^{-6} = 2^{8a}$

This step is crucial as it sets the stage for equating the exponents. By expressing both sides with the same base, we create a direct relationship between the exponents, allowing us to solve for the unknown variable. The transformation into a common base is a fundamental technique in solving exponential equations, ensuring that the exponents can be compared directly.

Step 2: Equate the exponents

Since the bases are the same, we can equate the exponents:

$-6 = 8a$

Now we have a simple linear equation that we can easily solve for 'a'. Equating the exponents is a direct consequence of having the same base on both sides of the equation, making it a valid and straightforward step in the solution process. This step eliminates the exponential aspect of the equation, transforming it into a more manageable algebraic form.

Step 3: Solve for $a$

Divide both sides by 8:

$a = \frac{-6}{8} = -\frac{3}{4}$

Thus, the correct solution is $a = -\frac{3}{4}$. This final step provides the accurate value of 'a' that satisfies the original exponential equation. The arithmetic is straightforward, involving a simple division, but its correctness hinges on the preceding steps being executed accurately. This result is the culmination of the correct application of exponential rules and algebraic manipulations.

Conclusion

The error in the provided solution occurred in Step 4 where the exponents were equated without ensuring that the bases were the same. The correct solution involves expressing both sides of the equation with a common base of 2 before equating the exponents. The accurate solution to the equation $\frac{1}{64} = 16^{2a}$ is $a = -\frac{3}{4}$.

Understanding and applying the properties of exponents correctly is crucial for solving exponential equations. This detailed analysis highlights the importance of each step in the solution process and the need for precision in mathematical manipulations. By identifying the error and providing the correct solution, we aim to enhance understanding and proficiency in solving exponential equations. This comprehensive explanation serves as a valuable resource for students and anyone seeking to master exponential equations.