Error Analysis And Correction Of Exponential Equation Solution

Jul 9, 2025 by ADMIN 63 views

Introduction

In this article, we will delve into a step-by-step analysis of a presented solution to an exponential equation. Our primary focus will be to pinpoint the exact location of the error within the provided steps. Mathematics, especially when dealing with exponents and equations, demands meticulous attention to detail. A small oversight can lead to a cascade of incorrect results. Therefore, our approach will not only identify the mistake but also provide a comprehensive explanation of the correct methodology to solve the equation. This exploration is crucial for anyone learning or teaching mathematics, as it reinforces the importance of accuracy and a solid understanding of fundamental principles. The goal is to transform a potentially frustrating error into a valuable learning experience, emphasizing the critical role of careful calculation and logical progression in solving mathematical problems.

Problem Statement

The original problem presents an equation involving exponents and asks to solve for the variable 'a'. Let's re-examine the given steps:

1.  1/64 = 16^(2a)
2.  4^(-3) = (2^4)^(2a)
3.  4^(-3) = 2^(8a)
4.  -3 = 8a
5.  a = -3/8

At first glance, the solution appears to follow a logical progression, utilizing the properties of exponents. However, a closer examination reveals a critical error that invalidates the final result. Our task is to dissect each step, identify the flaw, and then demonstrate the correct way to arrive at the solution. This process will not only rectify the specific error in this example but also highlight common pitfalls in solving exponential equations, thus promoting a deeper understanding of the underlying mathematical concepts. By meticulously analyzing each step, we aim to equip readers with the necessary tools to confidently tackle similar problems in the future.

Identifying the Error

The error lies in the transition between step 2 and step 3. While the conversion of 1/64 to 4^(-3) and 16 to 2^4 is correct, the equation is not fully expressed in terms of the same base before equating the exponents. In the provided solution, the left side of the equation remains as 4^(-3), while the right side is correctly transformed to 2^(8a). To accurately solve the equation, both sides must be expressed with the same base. This step is crucial in simplifying exponential equations, as it allows us to directly compare the exponents once the bases are uniform. Neglecting to express both sides with the same base leads to an incorrect equation and, consequently, a flawed solution. Therefore, the error stems from a premature comparison of exponents without ensuring a common base across the equation.

To elaborate further, let's break down why this is a problem. The equation 4^(-3) = 2^(8a) has different bases on each side. You cannot directly equate the exponents (-3 and 8a) unless the bases are the same. The base 4 can be expressed as 2^2, which is a necessary step to align the bases and proceed with solving for 'a'. This oversight is a common mistake in solving exponential equations, emphasizing the importance of ensuring a uniform base before proceeding with further calculations. By pinpointing this error, we can now focus on the correct approach to solve the equation.

Correcting the Solution

To correctly solve the equation, we need to express both sides with the same base. Let's start from the beginning:

1.  1/64 = 16^(2a)

First, we can express both 1/64 and 16 as powers of 2:

1.  2^(-6) = (2^4)^(2a)

Here, we've rewritten 1/64 as 2^(-6) because 2^6 equals 64, and the reciprocal is indicated by the negative exponent. Similarly, 16 is expressed as 2^4. This step is crucial as it sets the stage for equating exponents, which is only valid when the bases are the same. By expressing both sides of the equation in terms of the base 2, we are ensuring a consistent foundation for the subsequent steps in solving for 'a'.

Next, simplify the right side using the power of a power rule:

2.  2^(-6) = 2^(8a)

Now that both sides of the equation have the same base (2), we can equate the exponents:

3.  -6 = 8a

This step is the core of solving exponential equations once a common base is established. By equating the exponents, we transform the problem into a simple algebraic equation. This direct comparison is only valid because we have ensured that the exponential expressions on both sides are based on the same number, allowing us to isolate the variable 'a'.

Finally, solve for 'a':

4.  a = -6/8 = -3/4

Therefore, the correct answer is a = -3/4. This result differs significantly from the incorrect answer obtained earlier, highlighting the importance of the proper steps in solving exponential equations. By carefully expressing both sides with a common base and then equating the exponents, we arrive at the accurate solution. This methodical approach not only provides the correct answer but also reinforces the fundamental principles of exponential equations.

Step-by-Step Breakdown of the Correct Solution

To further clarify the correct solution, let's break down each step with detailed explanations:

Step 1: Express both sides of the equation with the same base.

The original equation is 1/64 = 16^(2a). To solve this, we need to express both 1/64 and 16 as powers of the same base. The base 2 is a suitable choice because both 64 and 16 are powers of 2. This step is critical because it allows us to apply the fundamental property of exponential equations, which states that if b^x = b^y, then x = y. Without a common base, this property cannot be used directly, making it impossible to isolate the variable 'a'.

1/64 can be written as 2^(-6) since 2^6 = 64. The negative exponent indicates that we are dealing with the reciprocal of 2^6. Similarly, 16 can be written as 2^4 because 2^4 = 16. By rewriting both sides of the equation in terms of the base 2, we create a uniform structure that allows for direct comparison of exponents. This initial transformation is the foundation for solving the equation correctly.

Step 2: Rewrite the equation with the common base.

Substituting these expressions into the original equation, we get 2^(-6) = (2⁴⁾(2a). This step is a direct application of the base conversions we performed in the previous step. By replacing 1/64 and 16 with their equivalent expressions in base 2, we are effectively setting up the equation for the next simplification, which involves the power of a power rule. The clarity of this step is crucial for avoiding errors, as it directly translates the initial problem into a form that is easier to manipulate and solve.

Step 3: Apply the power of a power rule.

The power of a power rule states that (b^m)n = b^(m*n). Applying this rule to the right side of the equation (2⁴⁾(2a), we multiply the exponents 4 and 2a to get 8a. This simplification is a key step in reducing the complexity of the equation. The result is 2^(-6) = 2^(8a). This equation is now in a form where the bases are the same, and we have a single exponent on each side, making it possible to equate the exponents and solve for 'a'.

Step 4: Equate the exponents.

Now that both sides of the equation have the same base (2), we can equate the exponents: -6 = 8a. This step is the logical culmination of the previous transformations. The property that allows us to equate exponents is a cornerstone of solving exponential equations. By recognizing that the bases are equal, we can shift our focus from the exponential expressions to the exponents themselves, converting the problem into a simple algebraic equation. This transition is a powerful technique for simplifying complex mathematical problems.

Step 5: Solve for 'a'.

To solve for 'a', we divide both sides of the equation -6 = 8a by 8: a = -6/8. This is a straightforward algebraic manipulation. By isolating 'a' on one side of the equation, we determine its value. However, it is important to simplify the fraction -6/8 to its simplest form, which is -3/4. This final simplification is crucial for presenting the answer in its most concise and accurate form. Therefore, the correct solution is a = -3/4.

Conclusion

In conclusion, the initial solution presented contained an error in prematurely equating exponents without ensuring a common base. The correct approach involves expressing all terms with the same base, applying exponent rules, and then equating the exponents to solve for the variable. The correct solution for the equation 1/64 = 16^(2a) is a = -3/4. This exercise underscores the importance of a methodical and precise approach when working with exponential equations. By carefully following each step and ensuring that all conditions are met, one can avoid common pitfalls and arrive at the correct answer. This detailed analysis not only corrects the specific error in this problem but also provides a valuable framework for tackling similar mathematical challenges in the future.

Keywords

Exponential equations, error analysis, correct solution, exponents, common base, power of a power rule, solving for variables, mathematical errors, step-by-step solution, equation simplification. These keywords are carefully chosen to encompass the main topics discussed in this article, enhancing its discoverability and relevance for readers seeking information on these subjects. They also serve to reinforce the key concepts and methodologies presented, ensuring that the core themes of the article are clearly communicated and easily accessible.