Maude's Evaluation Of (3²)⁻³ Analysis And Solution

#h1

Maude attempted to evaluate the expression (3²)⁻³. Her steps and the final result have led to a discussion about the correct application of exponent rules. Exponents, especially when nested, require careful attention to the order of operations and the specific rules that govern their manipulation. This article delves into Maude's solution, dissecting each step to determine its validity. We'll explore the power of a power rule, the correct procedures for managing exponents, and highlight common pitfalls to avoid. By the end, you'll have a clearer understanding of how to approach similar exponential expressions and confidently solve them.

Maude's Solution

#h2

Maude's evaluation of the expression (3²)⁻³ involved the following steps:

1. 3^(2 * -3)
2. 3^6
3. 729

To assess the correctness of Maude's work, we need to meticulously examine each step. The first step involves applying the power of a power rule, which states that (a^m)n = a^(m*n). Maude correctly identified this rule and initiated the solution by multiplying the exponents. However, there is a subtle yet critical error in the subsequent steps that significantly alters the outcome. The key to solving exponential expressions lies in the proper application of these rules, and any deviation can lead to an incorrect final answer. In the following sections, we'll scrutinize each step, pinpointing where Maude's solution veered off course and highlighting the correct path to the solution.

Analyzing Step 1: 3^(2 * -3)

#h2

The first step in Maude's solution, 3^(2 * -3), is where the power of a power rule is applied. This rule, a cornerstone of exponent manipulation, states that when raising a power to another power, you multiply the exponents. In this case, the base is 3, the inner exponent is 2, and the outer exponent is -3. Applying the rule, we multiply 2 and -3 to get -6. Therefore, the expression becomes 3^(-6). Maude correctly identifies the need to multiply the exponents, which demonstrates an understanding of the underlying principle. However, the subsequent execution of this principle is where the error arises. This initial step is crucial because it sets the foundation for the entire solution. A mistake here will inevitably propagate through the remaining steps, leading to an incorrect result. The power of a power rule is fundamental in simplifying complex exponential expressions, and its proper application is essential for achieving accurate solutions. Understanding the mathematical principles behind the rules will ensure correct application. The step 3^(2 * -3) is the correct initial transformation, but the next step needs careful examination. It’s not merely about applying the rule but also about correctly performing the arithmetic and interpreting the result. Maude's intent to use the power of a power rule is accurate, but the devil is in the details of the execution. To further understand this step, let's consider other examples of the power of a power rule in action, such as (2³⁾2 = 2^(32) = 2^6 or (5^-1)4 = 5^(-14) = 5^-4. These examples reinforce the principle of multiplying exponents when a power is raised to another power. This foundational understanding is essential before proceeding to the next step.

Identifying the Mistake in Step 2

#h2

Step 2 in Maude's solution is 3^6. Here's where the critical error occurs. While Maude correctly initiated the process of multiplying the exponents in the previous step, she made a mistake in the arithmetic. Multiplying 2 by -3 results in -6, not 6. Therefore, after applying the power of a power rule, the expression should be 3^(-6), not 3^6. This seemingly small arithmetic error has a significant impact on the final result. A positive exponent indicates repeated multiplication of the base, while a negative exponent indicates the reciprocal of the base raised to the positive exponent. The negative exponent is a key concept to grasp. The incorrect exponent in this step will lead to a completely different value. The mistake highlights the importance of meticulous attention to detail when working with exponents. It's not enough to understand the rules; you must also perform the calculations accurately. A simple sign error can throw off the entire solution. To avoid such mistakes, it's helpful to double-check each step and use techniques like writing out the multiplication explicitly (e.g., 2 * -3 = -6) to minimize the chance of error. This step underscores the fundamental nature of exponents and their sensitivity to sign changes. Getting the sign of the exponent correct is crucial for arriving at the correct final answer. Maude's error in this step emphasizes that even with a solid understanding of the rules, a minor arithmetic misstep can lead to a wrong conclusion. Therefore, careful calculation is as important as the correct application of the rules.

Correcting the Solution

#h2

To correct Maude's solution, we need to backtrack to the point of error and proceed with the correct calculations. As we identified, the mistake occurred in Step 2, where Maude incorrectly stated the expression as 3^6 instead of 3^(-6). The correct progression from Step 1, which was 3^(2 * -3), is as follows:

1. 3^(2 * -3)
2. 3^(-6) // Corrected step

Now, we need to evaluate 3^(-6). A negative exponent indicates the reciprocal of the base raised to the positive exponent. In other words, a^(-n) = 1/a^n. Applying this rule, we can rewrite 3^(-6) as 1/(3^6). To complete the evaluation, we need to calculate 3^6, which is 3 * 3 * 3 * 3 * 3 * 3 = 729. Therefore, 1/(3^6) = 1/729. The corrected solution demonstrates the importance of understanding negative exponents and their relationship to reciprocals. It also highlights the need to apply the correct exponent rules consistently. The process of correcting the solution not only provides the right answer but also reinforces the principles of exponent manipulation. By identifying and rectifying the error, we gain a deeper understanding of the underlying mathematical concepts. This step-by-step correction serves as a valuable learning experience, illustrating how a single mistake can alter the outcome and emphasizing the importance of careful and accurate calculations. The corrected solution shows the power of understanding and correctly applying exponent rules to arrive at the accurate answer. This meticulous approach ensures the integrity of the mathematical process and the reliability of the result.

Final Answer and Explanation

#h2

The correct evaluation of the expression (3²)⁻³ is 1/729. Maude's error was in Step 2, where she incorrectly calculated 3^(2 * -3) as 3^6 instead of 3^(-6). This seemingly minor arithmetic mistake led to a significantly different final result. The negative exponent is crucial in determining the correct answer. A negative exponent indicates the reciprocal of the base raised to the positive exponent. Therefore, 3^(-6) is equivalent to 1/(3^6). To arrive at the final answer, we first calculate 3^6, which is 729, and then take its reciprocal, resulting in 1/729. The final answer, 1/729, stands in stark contrast to Maude's answer of 729, highlighting the impact of the initial error. The explanation underscores the significance of precision in mathematical calculations and the importance of understanding the properties of exponents, particularly negative exponents. The solution serves as a valuable learning opportunity, emphasizing the need for careful attention to detail and thorough understanding of mathematical principles. By understanding the correct application of exponent rules and the meaning of negative exponents, we can confidently solve similar problems and avoid common pitfalls. The entire process, from identifying the error to correcting the solution and arriving at the final answer, illustrates the importance of methodical problem-solving in mathematics. Each step builds upon the previous one, and a single mistake can have cascading effects on the final result. The final answer, 1/729, represents the culmination of the corrected solution, demonstrating the power of accurate calculations and a deep understanding of exponential expressions.

Is Maude Correct?

#h2

No, Maude is not correct. Her mistake lies in the arithmetic calculation of the exponent in Step 2. She incorrectly stated that 3^(2 * -3) is equal to 3^6, when it should be 3^(-6). This error stems from multiplying 2 and -3 and incorrectly getting a positive 6 instead of a negative 6. As a result, her final answer is incorrect. Maude's incorrect result underscores the importance of paying close attention to the signs of exponents. A negative exponent drastically changes the value of the expression, as it represents the reciprocal of the base raised to the positive exponent. Maude’s initial application of the power of a power rule was correct in principle, but the arithmetic error in determining the sign of the resulting exponent led to the incorrect solution. The mistake highlights a common pitfall in exponent manipulation and serves as a valuable learning experience. To avoid such errors, it is essential to double-check all calculations, especially when dealing with negative numbers. The correct solution involves recognizing that 3^(-6) is equivalent to 1/(3^6), which ultimately leads to the correct answer of 1/729. Maude's error emphasizes that even a small arithmetic mistake can have significant consequences in mathematical calculations, particularly when dealing with exponents. Therefore, careful attention to detail and a thorough understanding of the rules are crucial for achieving accurate results. The discrepancy between Maude's answer and the correct answer underscores the need for meticulousness in mathematical problem-solving.

Correct Answer Choice

#h2

Based on our analysis, the correct answer choice is neither A nor B, as neither option accurately reflects Maude's error. Option A, "Yes, she is correct," is clearly incorrect since we have identified a significant arithmetic error in her solution. Option B, "No, she should have added the exponents instead of multiplying," is also incorrect because Maude correctly applied the power of a power rule, which involves multiplying the exponents. The issue was not the rule she used but rather the arithmetic error in the calculation. A more appropriate answer choice would explicitly state that Maude made a mistake in calculating the exponent, resulting in an incorrect final answer. The correct approach involves multiplying the exponents (2 and -3) to get -6, resulting in the expression 3^(-6). This expression then needs to be interpreted as the reciprocal of 3^6, which is 1/729. The absence of a suitable option in the given choices highlights the importance of a comprehensive understanding of the problem and the ability to identify and articulate the specific error made. The correct explanation of Maude's error focuses on the arithmetic mistake in calculating the exponent and emphasizes the significance of negative exponents in the final result. The analysis demonstrates the need for a clear and accurate representation of the error in the answer choices to effectively assess understanding and problem-solving skills. Therefore, it’s important to not only understand the correct solution but also to accurately pinpoint the specific mistake that led to the incorrect answer.

Conclusion

#h2

In conclusion, Maude's evaluation of the expression (3²)⁻³ provides a valuable learning opportunity for understanding exponent rules and the importance of accurate calculations. While Maude correctly applied the power of a power rule initially, she made an arithmetic error in Step 2, leading to an incorrect final answer. The key takeaway is the significance of meticulousness in mathematical problem-solving. Even a small arithmetic mistake, such as miscalculating the sign of an exponent, can have a substantial impact on the result. The correct evaluation involves multiplying the exponents 2 and -3 to get -6, resulting in 3^(-6). This expression is then correctly interpreted as the reciprocal of 3^6, which is 1/729. The correct final answer highlights the critical role of negative exponents and their relationship to reciprocals. By carefully analyzing each step of Maude's solution, we were able to pinpoint the specific error and correct it, reinforcing the principles of exponent manipulation. This exercise demonstrates the importance of not only understanding the rules but also applying them accurately and consistently. The analysis underscores the need for careful attention to detail and thorough understanding of mathematical concepts to achieve correct solutions. The learning derived from Maude's mistake serves as a reminder that precision and accuracy are paramount in mathematical problem-solving. By understanding these principles, we can confidently tackle similar exponential expressions and avoid common pitfalls.