Students’ Misconceptions About Exponents

Abstract

This study analyzes student misconceptions about exponents, an essential element of the middle school mathematics curriculum. Content analysis, a qualitative research methodology, was used. The study group was 150 students attending a public middle school in Kayseri, Turkey. A specially designed, formulated eight-question exponents conceptual test was administered as a data collection tool. The exponents test was revised with expert opinions to identify students’ misconceptions regarding exponents. The final version of the exponents conceptual test includes eight questions. In the analysis, the students’ answers were individually classified as correct, incorrect, or blanks, and the specific misconceptions they held, along with examples of correct and incorrect responses (as misconceptions), were analyzed. Students exhibit several misconceptions regarding exponents, including an inability to calculate exponents, a misunderstanding of the condition of zero exponent, and a failure to distinguish between different exponent cases, such as −a^b versus (−a)^b. They also struggle to comprehend negative exponents, fail to recognize that even exponents yield positive values, and confuse the relationships between a^b and b^a. These challenges reflect broader conceptual misunderstandings related to the operational properties of exponents.

Keywords

concept exponents middle school misconception irrational number

Introduction

Meaningful learning occurs when learners can link newly acquired information to their existing knowledge (Ausubel, 1960). Understanding the reasons behind one’s actions is called relational understanding (Skemp, 1976). This approach emphasizes the study of how students learn and develop mathematical concepts (Tall & Vinner, 1981). Students who lack relational understanding in mathematics may experience misconceptions or challenges concerning several subjects. Dependence on memorizing explanations or rules, instead of fostering meaningful learning, creates a disconnection between procedural and conceptual knowledge. This disconnection can hinder students’ ability to learn effectively and foster the emergence of misconceptions (National Council of Teachers of Mathematics, 2005).

Mathematics teachers employ various strategies for explaining concepts to their students and examine the effects of these strategies (Hiebert & Carpenter, 1992). In the learning and application of mathematical knowledge, understanding concepts is crucial. Given that mathematics comprises numerous concepts that build upon one another, students may make mistakes and develop conceptual misunderstandings. At every stage of mathematics education, the content consists of abstract concepts that are often difficult for students to relate to real-life situations. This disconnect can lead to misunderstandings (Duatepe-Paksu, 2008).

If students fail to achieve meaningful learning—meaning they cannot connect what they learn to their daily lives—they may resort to rote memorization and struggle to transition between different pieces of information (Jackson & Cobb, 2010). Rote learning is particularly susceptible to conceptual misunderstandings because it encourages students to memorize information, especially rules, without fully understanding the underlying concepts. As a result, students may not be adequately prepared for subsequent abstract topics, which can negatively impact their ability to engage in meaningful learning (Simon et al., 2004).

Students need to achieve meaningful learning when transitioning from basic operations to exponents. They frequently attempt to apply the principles acquired with integers to exponential numbers. However, based solely on their experiences with integers, students generally assume that exponentiation simply means multiplication. The assumption leads to confusion, particularly when they encounter expressions like raising the number x to the power of 1/2. They struggle to understand the idea of “the product of half of x with itself” (Ben-Hur, 2006; Tall, 2002).

Exponents are connected to early algebra topics and early calculus topics that students will encounter in later stages of learning. From this viewpoint, the misconceptions students encounter may appear specific to a particular topic or question, but they actually impact numerous interrelated topics and other subjects, for example, chemistry and physics. Given that the eighth-grade mathematics curriculum includes topics such as negative exponents, the properties of exponential functions, and exponential expressions, we have chosen to focus our research on these students. This research aims to identify the misconceptions that eighth-grade students have regarding exponents and understand the reasons behind them. The research question is: “What are the misconceptions of eighth-grade students regarding exponential expressions?”

Theoretical Framework

Students have been quite successful in calculating exponents when both the base and the exponent are natural numbers. However, they frequently made mistakes when the base was a decimal number and the exponent was a natural number. They expanded all exponent cases to include situations where the base and exponent consist of natural numbers. Özdeş (2013) identified the errors and misconceptions of students regarding exponential numbers. Similarly, a 10-question test on exponential expressions was administered to 156 eighth-grade students to identify their misconceptions from the High School Transition Exam (Koçman, 2013). In addition to identifying the existing 17 misconceptions, 8 new ones were also identified. Then, the most common misconception about exponents were found to be mistakes made when calculating negative exponents, mistakes made when working with exponential expressions, mistakes made when writing numbers in scientific notation, and misunderstandings when writing numbers in scientific notation in the wrong order (Uçar, 2019). The misconceptions that students have about exponents are connected to the learning goals of “calculating the powers of integers, understanding the basic rules related to exponential expressions, and forming equivalent expressions” (Göçük, 2019). Students were asked to compare the two given exponents, indicate the result with the appropriate sign (greater than, less than or equal to: <, >, =), and write the justifications for their answers (Avcu, 2010).

One of the most common mistakes and misunderstandings students have about exponents is that they can’t tell the difference between “the square of a negative number” and “the negative of a squared number.” For example, many students believe that both (−4)² and −4² yield the same result of 16 (Orhun, 1998). Research has explored the misconceptions and calculation errors that plague students as they grapple with exponential expressions (Şenay, 2002). Alarmingly, significant challenges persist in students’ comprehension and ability to perform operations with exponents. Moreover, the learning struggle is particularly acute when it comes to negative exponents and calculating the values of exponents. Students often define –x⁻² as x², a misunderstanding rooted in the belief that “the product of two negative numbers is positive.” This misguided notion leads them to assume that a negative exponent combined with a negative base will somehow culminate in a positive outcome during calculations. Cengiz (2006) created an enlightening exponents test with 16 questions that assess understanding of exponents, along with 6 questions measuring computational skills, all aimed at shedding light on the misconceptions students have.

The “inability to grasp the significance of zero exponents” holds particular weight among the various misunderstandings surrounding exponents. It’s not uncommon for students to assert that x⁰ equals either x or 0. Similar prevalent misstep is their failure to differentiate between the expressions (-x)ⁿ and xⁿ, often leading them to erroneously multiply the exponent and the base while evaluating exponents. Moreover, many students frequently have misconception by performing direct multiplication or division between exponents when dealing with exponents.

This widespread confusion among students often arises from the frequently encountered statement in algebra textbooks: “If x² = 25, then x = ±5” (Roach et al., 2004). This phrase encapsulates the challenges students face and underscores the need for a more nuanced understanding of exponential concepts.

Students need to focus on repeated multiplication operations in exponential functions. In this context, Confrey and Smith (1994, 1995) noted that it is more appropriate to approach multiplication not as repeated addition but based on the primitive multiplication process they called “splitting.”“Splitting” is defined as the process of replacing an object with a fixed number of copies. In discrete cases, the division process can be visually represented with a tree diagram. Students will examine the patterns that arise from the comparisons between consecutive inputs and their corresponding consecutive outputs when it comes to tabular data. To explain exponential functions, they say that teaching them will be much more effective if they are taught with additive inputs (1, 2, 3…), multiplicative outputs (b1, b2, b3, etc.), and the idea of division. In the context of extending exponential functions from positive integers to rational numbers, they also discuss the concept of “interpolation in the worlds of counting and division.” The repeating operation, in the world of counting, transforms from being added in the form of 1/n to being multiplied in the form of b1/n in the world of division.

Confrey and Smith’s (1994) research has been a source of inspiration for many subsequent studies on the construction of exponential functions. They suggested starting with discrete exponential reasoning to build exponential functions and then moving on to continuous exponential reasoning as the values of exponentials become denser on the real number line. For example, Weber (2002) observed the necessity of helping students consider exponential functions to be a process. In this context, he used activities that allowed students to see b^x as a process consisting of x instances of b. According to Weber (2002), this approach enables students to perceive the exponential process as an object capable of performing operations. Students can observe this situation when they attempt to understand or reconstruct exponential rules.

Pitta-Pantazi et al. (2007) proposed a data-based model for the process of students performing exponential operations. They examined how students compare different exponents and justify their reasoning. Students primarily interpret the expression when a and x are positive integers and can extend this understanding to include positive fractional bases. At the second level of the model, students have extended this understanding to include fractional bases and integer exponents. They have understood the property a^xa^y = a^x + y in a process-based manner for positive integers a, x, and y. In the third and final level of the model, students reconstructed the meaning of the exponential operation and understood the expression ax in cases where a and x are rational numbers; they were able to express this understanding verbally (Pitta-Pantazi et al., 2007).

Researchers came up with activities that help students think of the exponential operation process as a process (Confrey & Smith, 1994, 1995; Ferrari-Escolá et al., 2016; Weber, 2002). The activities help students think of an additive input sequence (exponents: 1, 2…, n) and a multiplicative output sequence (powers of a specific base: b₁, b₂, b₃,…, bn…) together. Ferrari-Escolá et al. (2016) used sets of cards in the form of (n|y), where n is a positive integer and y = 2n, and asked students to determine missing cards or card sections, sort the cards, and perform an operation between card pairs to obtain another card from the card set as a result of this operation. Inspired by the historical development of logarithms, he suggested that students multiply the two numbers on the top of the cards (e.g., (n|y)), relate the values on the bottom of the cards to addition, and associate dividing the two values on the top of the cards with subtracting the values on the bottom. This process allows for an understanding of the rules of exponential and logarithmic functions. Additionally, by reversing the independent and dependent variables (the top or bottom parts of the cards), students can relate exponential functions to their logarithmic inverses. However, we have not used these materials to extend exponential functions to rational numbers or introduce logarithms.

Persistent errors and misconceptions students make while simplifying exponential expressions during the transition from algebra to calculus (Cangelosi et al., 2013). Classifying the persistent errors made by students highlighted the difficulties encountered during the manipulation of negative signs in exponents. Extending exponential functions from positive integer exponents to all integers will be challenging for many students (Cangelosi et al., 2013).

Levenson (2012) examined how well teachers understood the relationship and distinction between definition, proof, and theorem, focusing on exponential operations and the zero exponent in particular. Then, Kontorovich (2016) analyzed possible explanations for the mathematical rule b⁰ = 1. Researcher emphasized the importance of helping them understand that this rule is necessary for extending the exponent properties to include the zero exponent. The implementation of these suggestions can contribute to students developing a deeper understanding by moving the definition of zero exponent from being merely an isolated piece of memorized information (Kontorovich, 2016; Levenson, 2012).

In summary, misconceptions, such as incorrect learning and incorrect generalizations, are frequently encountered in exponential expressions. Here is a list of common misconceptions of exponents: Inability to determine the magnitude of numbers with huge or minimal exponents, inability to understand the meaning of zero exponents, and inability to determine the negative exponent of a number. Moreover, students often struggle to distinguish between the expressions x^a and a^x, struggle to calculate even or odd powers of negative numbers, struggle to make mental estimations when comparing numbers with different bases and exponents, and face difficulties when performing the four basic arithmetic operations (addition, subtraction, multiplication, and division) with exponents (Avcu, 2010; Cangelosi et al., 2013; Duatepe-Paksu, 2008; Ulusoy, 2019; Weber, 2002). Examining the mistakes, misconceptions, and learning difficulties that students frequently encounter on the topic of exponents, Duatepe-Paksu (2008) has proposed various solutions. However, she has also made evaluations of the processing of the relevant topics in the primary and middle school mathematics curriculum. Additionally, before the topics of powers and roots are covered, any deficiencies students may have in fundamental subjects such as integers, rational numbers, the four basic operations, and absolute value should be completely addressed.

Limitations of the Study

The study was conducted with 150 students attending the eighth-grade in Kayseri during the 2022 to 2023 and 2023 to 2024 academic years. It was limited to data obtained from a exponents test developed to identify misconceptions exponents. It was assumed that the students participated in the study voluntarily, that their answers reflected their perceptions, and that they answered the questions individually and sincerely.

Method

Research Model

We have used content analysis, one of the qualitative research models, to present students’ existing knowledge as it is. In this model, it is important to accurately observe, identify, and determine the researched subject without altering it (Cresswell, 2013; Fraenkel et al., 1993). The content analysis describes the relationship between situations by considering the connections between events and previous events, and detailing what individuals, groups, institutions, objects, and entities are. Additionally, in content analysis, it is necessary to thoroughly define and explain the given situation within the framework of certain standards to illuminate it. Given this characteristic, we expect descriptive studies to illuminate subsequent research by accurately identifying the situation. Observing and identifying things, writing them down, finding connections between events, and drawing broad conclusions are important parts of the research done in this model (Cresswell, 2013; Yıldırım & Şimşek, 2021). To prevent incorrect and incomplete learning, it is important to accurately identify the current situation.

Participants

The study group was chosen using the purposive sampling method. Purposive sampling enables thorough and detailed research. This method is employed when the research meets the criteria, exhibits specific traits, and encompasses several exceptional conditions. The sampling method and sample size must be established according to learning outputs of each investigation. In scenarios when the sample size is rich in information, the validity, significance, and insights that researchers derive from their observational and analytical skills are paramount (Cresswell, 2013). The pilot application, representing the first part of the research, was conducted with 30 randomly selected eighth-grade students from a middle school in Kayseri during the 2022 to 2023 academic year. The actual research group consists of 150 eighth-grade students attending various middle schools in the same region of Kayseri and with the same socioeconomic level during the 2022 to 2023 and 2023 to 2024 academic years. All the schools are public. Schools in the same area that the researcher could easily access were randomly selected. The researcher chose the study group of eighth-grade students because the eighth-grade-level middle school students have a better grasp of the topic of exponential numbers, and in the Ministry of National Education (MoNE) eighth-grade learning outputs, the topic of square roots follows exponential numbers.

Data Collection Tools

Turkish Ministry of National Education (MoNE, 2018) curriculum used the development of the conceptual test of exponents. Table 1 displays the learning outcomes taken into account when preparing the exponent questions for the conceptual test. An exponents test for eighth-grade students was developed and was administered it to 30 students. However, due to the large number of questions and the researcher asking students for detailed explanations, the students could not look at every question, and results suitable for the research could not be obtained. Therefore, the researcher gathered the necessary expert opinions and developed a new version of the exponents test. In the new exponents test, there are eight questions on exponents (with sub-items under question eight). Table 2 provides the types of misconceptions present in exponent questions and identifies which type each exponent question belongs to. We have provided an answer key for the Akdemir Kabalcı (2025) exponents conceptual test. Research on exponential number misconceptions, eighth-grade learning outputs, and resources from the Turkish Ministry of National Education (MoNE) that were related to the subject (EBA–education information network of MoNE, textbooks, etc.) were the main things that were looked at when the test to determine exponent misconceptions was being made. Therefore, we have included questions that are appropriate for the students’ respective levels. Before the application began, the researcher asked the students to write down their solution methods and explanations while solving the questions. The purpose here is to determine which conceptual misunderstanding the student fell into if they made a mistake. Students received one class hour (40 min) during the test administration. Akdemir Kabalcı (2025) included the Exponential Conceptual Test. The findings are presented on a question-by-question basis from the conceptual test of exponents.

Table 1.

The Learning Outputs Related to Exponents Questions.

Code	Learning outputs
M.8.1.2.1.	“Calculates the powers of integers.” (MoNE, 2018, pp. 71)
M.8.1.2.2.	“Understands the basic rules related to exponential expressions, creates equivalent expressions.” (MoNE, 2018, p. 71)

Table 2.

Types of Misconceptions (M) in Exponents and the Distribution of Exponents Questions According to Types of Misconceptions (Table of Specifications).

Code	Explanation	Question
M1	Inability to distinguish between the -a^b and -a^-b cases	3
M2	Inability to distinguish between a^b and b^a situations (here, a represents a real variable; b represents a natural number)	5
M3	Inability to detect the negative exponent condition	4
M4	Inability to determine the zero exponent condition	2
M5	Inability to perceive the value as always positive when the exponent is even.	6
M6	Difficulty with operational properties in exponential expressions	8
M7	Inability to calculate the value of an exponentiation	1
M8	Difficulty in calculating the exponent of a power number	7

Data Analysis

One point was given for each correct, and zero to misconception, and blank answer on the exponent misconception test that was made to find out what the students didn’t understand about exponential numbers. We determined the total number of correct, misconception, and blank answers for each data point in this manner. We asked students to explain how they solved each question during the implementation phase of the exponents misconception test. The aim here is to determine whether the student has fallen into a misconception related to the topic and, if so, what type it is. After the data was obtained, it was processed into Microsoft Excel. The students’ responses to the misconception identification test were initially examined by the first researcher. The questions were categorized based on misconceptions. In the initial review, it was checked whether the answers were consistent with the table of specification given in Table 2. Afterward, the first researcher independently evaluated each response and recorded it in an Excel file. Afterward, the second researcher also independently evaluated each answer and recorded the scores and misconceptions in an Excel file. Afterward, the scores and consistency of the two researchers were compared using Excel. The researchers’ scores overlap 96%. There is consensus on the remaining 4%.

Validity and Reliability

We created the exponents misconception test by examining the examples, exercises, and unit assessment questions on the MoNE website and in eighth-grade mathematics textbooks. We created it in this manner to ensure internal validity. The purpose of creating the exponents misconception test is to determine whether students fall into misconceptions when they encounter exponents. The reliability and validity of the misconception test on exponentials were determined by calculating the test’s internal consistency coefficient using Cronbach’s alpha.

The data obtained from the test prepared to reveal the misconceptions experienced by students on exponentials, is presented. Based on the students’ answers to the questions, the number of correct, misconception, and blank responses was calculated. The results obtained from the data are shown Figure 2 on pie and Figure 3 on bar graphs. The data obtained from the misconception test were analyzed using the SPSS analysis program. Based on the students’ answers, misconception questions or statements that exemplify conceptual misunderstandings in their explanations, even if the answer is correct, have been recorded. Validity analysis are presented under the subheadings exploratory factor analysis (EFA).

Exploratory Factor Analysis of Exponents Misconception Test

There are two important conditions for exploratory factor analysis: The Kaiser-Meyer-Olkin (KMO) value and the results of Bartlett’s test. The KMO test helps us determine if the sample size is sufficient. If the KMO value is greater than 0.7, it indicates that the study group has established a good relationship. The KMO value for exponential expressions was found to be 0.703 after analyzing the results of the success test. This value indicates that the number of working groups is sufficient. According to Bartlett’s test data, p < .00 (χ² = 141.397, df = 36), indicating that the test was significant (Pallant, 2020; Tabachnick & Fidell, 2007). These results indicate that they are consistent with the study group and the EFA, which are indicators of data distribution normality.

To determine the number of factors in the test, the eigenvalues of the factors must be greater than 1 and account for 5% of the total variance (Seçer, 2013). The total variance values for the exponential expressions belonging to the exponents test are given in Table 3. As given in Table 3, there are two factors with an eigenvalue greater than one and explaining at least 5% of the total variance. The total variance of these two factors was measured at 37.47%.

Table 3.

Total Variance Explained for Exponential Misconception Test.

Items	Initial eigenvalues			Extraction sums of squared loadings			Rotation sums of squared loadings
Items	Total	Variance (%)	Cumulative (%)	Total	Variance (%)	Cumulative (%)	Total	Variance (%)	Cumulative (%)
1	3.075	38.435	38.435	2.520	31.498	31.498	1.566	19.571	19.571
2	1.083	13.539	51.974	0.478	5.972	37.470	1.432	17.899	37.470
3	0.910	11.372	63.345
4	0.855	10.692	74.038
5	0.639	7.984	82.022
6	0.546	6.822	88.844
7	0.475	5.939	94.783
8	0.417	5.217	100.000

Another way to determine the number of factors in the test is to use a scree plot (Seçer, 2013). The scree plot for the exponents is given in Figure 1. It represents a factor between the points. The lines flatten out after a certain point. The contribution of factors after the point where the flattening begins to the variance is quite small (Çokluk et al., 2012). According to the scree test, the lines begin to flatten after the second factor. Based on this, the test is two-factor.

Figure 1.

Scree plot of answers on the exponents test.

After determining the number of factors in the test, it is necessary to determine which questions are included in the factors and the factor loading values of the questions. The factor loading values of the questions should be greater than 0.30 (Seçer, 2013). The factor loading values for the questions about the exponents from the test are given in Table 4. The factor loading values of the questions are composed of two factors. The first factor consists of questions 4, 5, and 8. The second factor consists of questions 1, 2, 3, 6, and 7. Looking at the content of the questions in the factors, the questions in the first factor are related to “M.8.1.2.2. Understands the basic rules related to exponential expressions and creates equivalent expressions. (MoNE, 2018, p. 71) was found to belong to the learning output, and it was aimed more at measuring students’ conceptual knowledge and interpretation skills. The questions in the second factor, “M.8.1.2.1. Calculates integer powers of integers.” (MoNE, 2018, p. 71), were found to belong to this learning output. It was more aimed at measuring students’ procedural knowledge about exponents.

Table 4.

Factor loadings for the exponents test.

Question	Factor 1	Factor 2
Exponent_8	0.670
Exponent_5	0.606
Exponent_4	0.578
Exponent_7	0.535	0.491
Exponent_6		0.708
Exponent_1		0.585
Exponent_2		0.302
Exponent_3		0.247

Findings

The answers of the students to the exponential test were examined, and the descriptive values obtained in terms of the number of correct, misconception, and empty answers are given in Table 5. To perform a more detailed analysis of the test results, the existing results were first analyzed using a pie chart, and the graphs are presented in Figures 2 and 3. Upon examining the graphs, the percentage of misconceptions among the eight questions students answered about exponential expressions generally ranged from 10% to 36%. The lowest percentage of misconceptions, 10% (M = 89; SD = 0.302), was in question 1, while the highest percentage of misconceptions, 36% (M = 54; SD = 0.446), was in question 8. However, students’ percentage distribution of unanswered questions ranged from 8% to 29%. The first question was empty with a value of 8%, and the fifth question was empty with a value of 29% (M = 58; SD = 0.421). When evaluating the correct answers, the lowest correct answer rate was 41% for question 5, while the highest correct answer rate was 82% for question 1. However, the correct answer rates of students were 70% or higher in questions 1, 2, 6, and 7, which were considered successful. The collective representation of the answers to exponential expressions questions using a column graph is shown in Figure 3.

Table 5.

Students’ Descriptive Results on the Exponential Misconception Test.

Question	Correct	Misconception	Empty	M	SD	Analysis N	Missing N
1	123	15	12	0.89	0.302	148	10
2	112	20	18	0.85	0.340	148	16
3	82	44	24	0.65	0.441	148	22
4	84	49	17	0.63	0.459	148	15
5	62	45	43	0.58	0.421	148	41
6	114	22	14	0.84	0.354	148	12
7	105	27	18	0.80	0.382	148	16
8	64	54	32	0.54	0.446	148	30

Figure 2.

Answer distribution of questions in the exponential misconception test.

Figure 3.

Answer distribution of questions in the exponential misconception test.

The answers given by students to the exponents misconception test were examined, and examples from student answer sheets were provided for each question based on the data. The first question measures the learning output, “M.8.1.2.1. Calculates the powers of integers.” (MoNE, 2018, p. 71). 82% of the students answered correctly, 10% answered by misconception, and 8% left it blank. In Figure 4, the students who answered in a right manner understood what the exponents meant and correctly applied the exponent rule. Students who provide misconception answers have a misconception by confusing the exponent with the base. Another student who has experienced a misconception by multiplying the exponent by the base. Figure 4 illustrates that a student with a misconception committed an error during the multiplication operation. Students generally confuse the base with the power. As given in Figure 4, students use the power as a base and multiplied five times. Power used as base and base is used as power. Some students made mistakes in the calculation steps of their solutions on their answer sheets.

Figure 4.

Example of correct and misconception answers to the first question.

The second question measures the learning output: “M.8.1.2.1. Calculates the powers of integers.” (MoNE, 2018, p. 71). 75% of the students answered correctly, 13% answered incorrectly, and 12% left it blank. Figure 5 provides an example of the correct answer for the second question. The students correctly explained and wrote that the result was a 1. As given in Figure 5, some of them wrote as an informal proof of the rule. Some students indicated that the result was 1 by providing an example without any explanation. Among the students who gave incorrect answers, they thought the zero power of a number (except for 0) was obtained by multiplying the exponent by the base, as in the first question. Another student wrote an incorrect result without any explanation. Some students wrote that the result was equal to the number at the base. Figure 5 provides a misconception example. The student has over generalization about multiplying by zero. When students see the zero, they think of multiplication by zero, and the result is zero.

Figure 5.

Example of the correct and misconception answer to the second question.

The third question measures the learning output, “M.8.1.2.1. Calculates the powers of integers.” (MoNE, 2018, p. 71). Unlike the first and second questions, it aims to determine whether students understand whether a negative number is being raised to a power before the operation step or if a negative sign has been placed in front of the number being raised to a power. 55% of the students answered correctly, 29% answered incorrectly, and 16% left it blank. In Figure 6, the students who answered correctly emphasized the importance of parentheses when raising a negative number to an odd power. When there are no parentheses, they answered by noting that the negative sign in front of the exponent makes the result negative. Students who answered with a misconception, on the other hand, thought that the negative sign in front of the exponent belonged to the number when there were no parentheses, contrary to the students who answered correctly. In such types of questions, students who fall into misconceptions are likely to submit incorrect answers when they encounter an even power, regardless of whether there is a parenthesis or not. Another student was making an error in the order of operations. In other words, the student correctly expanded the exponent but incorrectly calculated the result of the multiplication. Figure 6 provides a misconception example. When we compare the correct answer with the misconception, students did not grasp the idea of the minus sign. They applied the general exponent rule to the −2³. The idea is to multiply the base by the power. The minus sign is also repeated. This is an overgeneralization of the basic rule of exponents. They could not differentiate the negative sign or negative numbers from the positive numbers.

Figure 6.

Example of the correct and misconception answer to the third question.

The fourth question measures the learning output: “M.8.1.2.1. Calculates the powers of integers.” (MoNE, 2018, p. 71). 56% of the students answered correctly, 33% answered incorrectly, and 11% left it blank. As shown in Figure 7, the students who answered correctly understood the meaning of positive and negative forces correctly. They have processed it and reached the correct result. Among the students who answered incorrectly, some misunderstood negative exponents by believing that the sign of the exponent applied to the base number, leading them to interpret the exponent as a negative number incorrectly. Students encountered a misconception when they multiplied the exponent by the base. Similar to the previous questions, students who correctly expanded the exponent of 3⁴, but they could not multiply by negative integers and negative integers as the power. Figure 7 shows a misconception example. Students used the negative sign of the power as a base. They understand using the negative sign while multiplying as a base, not as a power. They could not build a relation with the reverse operations or reciprocals. They could not differentiate the exponent operations, so they simply added these 2 exponents as 81 + 81. Probably, they could not differentiate the operations, and they could not write an explanation about the operations.

Figure 7.

Example of the correct and misconception answer to the fourth question.

The fifth question measures the learning output, “M.8.1.2.1. Calculates the powers of integers.” (MoNE, 2018, p. 71). 41% of the students answered correctly, 30% answered incorrectly, and 29% left it blank. The students who responded correctly expressed the expansion of the exponential expression appropriately when they saw the exponent as an integer, without being affected by the base being unknown. When they saw the base as an integer and the exponent as unknown, they wrote example numbers in place of the unknown in the exponent and correctly expressed the expansion of the resulting exponential expression. Additionally, they correctly understood that in the first expression, the square of any integer was taken, and in the other, the power of two to any integer was taken. An example of the correct answer is provided in Figure 8. When examining the solutions of the students who incorrectly solved the fifth question, the most important point observed is their inability to see that both of these expressions are exponents. In the first exponent, they correctly expanded x squared, but in the second one, they perceived two not as any integer exponent but as x + x, leading to a misconception. Other students mistakenly think these two exponents are the same, making it difficult to distinguish the base and exponent. Figure 8 provides a misconception example.

Figure 8.

Example of the correct and misconception answer to the fifth question.

The sixth question is related to the learning output: “M.8.1.2.1. Calculates the powers of integers.” (MoNE, 2018, p. 71). 76% of the students answered correctly, 15% answered incorrectly, and 9% left it blank. Figure 9 provides an example of the students who correctly answered. They solved the problem by determining whether to raise a negative number (in parentheses) to an even or odd power. They know that the sign of the exponents is positive when an even power is taken and negative when an odd power is taken. Therefore, they reached the correct answer. The student who gave the wrong answer misunderstood the concept by perceiving the number (−7) as adding it to themselves, resulting in an incorrect answer. Another student reached the result by multiplying the exponent and the base and experienced a conceptual misunderstanding. Another incorrect answer is one where the student correctly finds the numerical value of the exponential expression, regardless of the sign, but mistakenly writes the result as negative without paying attention to the fact that a negative number raised to an even exponent should be positive. Figure 9 provides a misconception example.

Figure 9.

Example of the correct and misconception answer to the sixth question.

The seventh question measures the learning output: “M.8.1.2.1. Calculates the powers of integers.” (MoNE, 2018, p. 71). 70% of the students answered correctly, 18% answered incorrectly, and 12% left it blank. When the answers of the students who answered correctly were examined, the students saw that an exponent was raised to another exponent; they first wrote the expansion of the expression inside the parentheses and then repeated this expansion outside the parentheses as many times as the exponent, reaching the final result. Another student who provided the correct answer encountered an exponent raised to an exponent and, by applying a similar method, realized that the exponents could be added in a shortcut manner. An example of the correct answer is provided in Figure 10. The student gave the wrong answer; upon seeing that another exponent of exponents was taken, they mistakenly thought that the exponents would be added, leading to a conceptual misunderstanding and incorrect operation. Some students were confused about the positions of the exponents. They made a mistake by performing operations according to the outer exponent first and then the inner exponent. Figure 10 provides an example of a misconception answer.

Figure 10.

Example of the correct and misconception answer to the seventh question.

The eighth question is aimed at measuring the learning output: “M.8.1.2.2. Understands the basic rules related to exponential expressions, creates equivalent expressions.” (MoNE, 2018, p. 71). 43% of the students answered correctly, 36% answered incorrectly, and 21% left it blank. Examining the papers with the correct answers showed that they understood the M.8.1.2.1 learning output, transitioned to M.8.1.2.2, and applied it correctly. They correctly applied the rules for how to expand an exponential expression and obtain equivalent exponential expressions. Figure 11 provides an example of the correct answer. When the students’ wrong answers were analyzed, they did not understand M.8.1.2.1 and could not move on to the M.8.1.2.2 learning output correctly, which led to misunderstandings. Figure 11 provides an example of a misconception answer.

Figure 11.

Example of the correct and misconception answer to the eighth question.

Summary of Exponents Misconceptions

A misconception of being unable to calculate the value of an exponent has been observed.

The misconception of being unable to determine the zero exponent condition has been observed.

Students experience the misconception of not being able to distinguish between the -a^b and (-a)^b cases.

It has revealed the misconception of students’ inability to perceive negative exponents.

Students experience a misconception of not being able to distinguish between the situations of a^b and b^a (where a represents a real variable and b represents a natural number).

When the exponent is even, students fail to perceive that the value is always positive, leading to misconceptions in their solutions.

The difficulty students have in calculating the exponent of a power has revealed a conceptual misunderstanding.

Students experience the misconception of struggling with the operational properties of exponential expressions.

Discussion

Students experience the misconception of being unable to calculate the value of an exponent. Some students have written the exponent as many times as the base and added them, while others have multiplied the exponent by the base (Cengiz, 2006; Crider, 1998; Orhun, 1998; Şenay, 2002; Soylu & Aydın, 2006; Yenen & Turhal, 2024). Some students have made conceptual errors by either multiplying or adding the exponent and the base, resulting in incorrect answers (Avcu, 2010; Birgin & Uçar, 2023; Duatepe-Paksu, 2008; Göçük, 2019; Güzel & Yılmaz, 2020; Özkan & Özkan, 2012; Uçar, 2019; Ulusoy, 2019; Yenen & Turhal, 2024; Yücesan, 2013). The reasons for the misconceptions encountered stemmed from procedural knowledge in some students and conceptual knowledge in others. If we elaborate a bit more on the procedural knowledge, the student has conceptualized what exponents mean in their mind, meaning they have reached a level where they can write the expansion, but they made an error in the repeated multiplication in the expansion they wrote, and found the result to be incorrect. Regarding conceptual knowledge, students could not comprehend the exponents in their minds, meaning they confused the concepts of exponent and base.

The inability to determine the zero exponent situation revealed a misconception. When students encounter the zero exponent, they mistakenly think, as in the first question, that they will multiply the base by the exponent and misinterpret the zero as the absorbing element in multiplication, resulting in zero. They added the exponent and the base, and they found the result equal to the number in the base (Soylu & Aydın, 2006; Cengiz, 2006; Crider, 1998; Duatepe-Paksu, 2008; Uçar, 2019; Yenen & Turhal, 2024). The reasons students fall for the misconception of the zero exponent have been observed to stem from their inability to fully comprehend the concept of exponential expressions. Due to this misconception, students think they will multiply the base and the exponent, and since zero is the absorbing element in multiplication, they find the value of the zeroth exponent to be zero. They write the base number as the value of the zero exponent because they add the exponent and the base, and since zero is the identity element in addition, they consider the zero exponent. They also switch the base and exponent and think they’re taking the zero exponent, which may explain their conceptual misunderstanding.

Students cannot distinguish between –a^b and (-a)^b. Students have experienced conceptual misunderstandings by thinking that taking the exponent of a negative number is the same as placing a negative sign in front of the expression whose exponent is being taken, leading to incorrect results in exponents. Sometimes students don’t realize that a negative integer base and an even exponent yield a positive result. A common mistake that students make is taking the wrong interpretation of the rule that says the sign of the result will be positive when a number is raised to an even power. It has been observed that some students, especially when encountering the third type of question, do not transition to the conceptual knowledge dimension regarding exponential expressions. They can’t tell the difference between odd and even powers of a negative number and can’t tell when parentheses are present or absent, which shows they haven’t moved on to the conceptual knowledge dimension (Avcu, 2010; Birgin & Uçar, 2023; Cengiz, 2006; Crider, 1998; Duatepe-Paksu, 2008; Orhun, 1998; Özkan & Özkan, 2012; Şenay, 2002; Yenen & Turhal, 2024; Yücesan, 2013).

Students have revealed the misconception that they cannot perceive the negative exponent situation. It is a common mistake to multiply the base and exponent numbers, swap the base and exponent numbers, ignore the negative exponent and treat it as a positive exponent, and think of the sign of the exponent as the sign of the base number (Birgin & Uçar, 2023; Cengiz, 2006; Duatepe-Paksu, 2008; Göçük, 2019; Soylu & Aydın, 2006; Uçar, 2019; Yenen & Turhal, 2024). Some students experienced a misconception, thinking that the values of both exponential expressions in the question were equal. The students’ answers shed light on the multiple causes of this mistake. According to the definition of exponents, students said they multiply the base number. However, when they generalized this rule to negative exponents, the idea of writing and multiplying the base number by a negative number seemed meaningless to the student. Therefore, they treated the power expression exponent as positive instead of negative and calculated its value. Another student with a misconception, for this reason, incorrectly answered by taking the exponent of a negative number, thinking that the negative sign of the exponent was like the sign of the base number. Some students, on the other hand, thought that a negative exponent resulted in a negative number.

Students experience the misconception of not being able to distinguish between the situations a^b and b^a (where a represents a real variable and b represents a natural number). Some students think both expressions are equal to the same value. Student conceptual misunderstandings were noted when they correctly wrote the first expression in the question, but mistakenly believed they should multiply the base and exponent in the second expression, or represent the first expression by adding it together as many times as indicated by the base number. The reason students experience these misconceptions is that they cannot fully comprehend the meaning of the exponential expression in their minds (Crider, 1998; Duatepe-Paksu, 2008).

The student solved problems without realizing that the value is always positive when the exponent is even. The situation of the exponent being an odd or even integer becomes important when the base is a negative integer. However, some students have ignored this situation and proceeded with the solution. In other words, they thought that if the base is negative, the result would also be negative, regardless of the exponent (Avcu, 2010; Birgin & Uçar, 2023; Cengiz, 2006; Crider, 1998; Duatepe-Paksu, 2008; Göçük, 2019; Özkan & Özkan, 2012; Şenay, 2002; Ulusoy, 2019; Yücesan, 2013).

The difficulty students have in calculating the exponent of a power number has revealed a conceptual misunderstanding. Students thought that when they encountered the exponent of a power, they needed to add the exponents. They generalized the rule that when exponential expressions with the same base are multiplied, the exponents are added, leading to a misconception. Some students experiencing misconceptions hold the belief that they should operate by repeatedly multiplying the exponent within the parentheses (Crider, 1998; Duatepe-Paksu, 2008).

Students often struggle with understanding the operational properties of exponents. Students experience multiple misconceptions simultaneously because the question about the properties of exponential expressions is a multiple-choice question. Students have experienced difficulty in determining the correct approach when performing four operations with exponential expressions. He thought that he could apply a rule he had learned or heard in every exponential expression, ignoring the type of operation—whether the numbers in the base and exponent were the same or different. For example, while multiplying two exponents with the same base, the rule of adding the exponents was observed; when multiplying exponential expressions with different bases, it was noted that both the base numbers and the exponents were multiplied. When they saw two exponential expressions with the same bases but different exponents, they added or multiplied the bases and exponents among themselves to obtain a new exponential expression. When multiplying exponential expressions with the same exponents but different bases, they mistakenly add the bases and write them with a common exponent. When it comes to division, it has been observed that students, upon seeing two exponential expressions with different bases being divided, divide both the base numbers and the exponents. In operations where two exponents with the same base but different exponents are divided, students subtract or divide the bases and exponents among themselves (Birgin & Uçar, 2023; Cancan et al., 2018; Cengiz, 2006; Crider, 1998; Duatepe-Paksu, 2008; Ulusoy, 2019). When they had to add or subtract with exponents, they used the rule that exponential expressions with the same base can be multiplied or divided to add or subtract. When the exponents are the same, they either apply this generalization to the base numbers to write them with a common exponent or create new exponential expressions by performing operations on both the exponent and the base numbers. When you add the same exponents together, if there is a coefficient in front of them, they multiply the coefficient by the base number to get new exponents. They then add the bases of those exponential expressions and write them under a common exponent, which gives you answers that don’t make sense (Cengiz, 2006; Duatepe-Paksu, 2008).

Students directly performed calculations on the exponent questions, stating in their explanations that they memorized the rule when they first started the subject (Sevgi & Akdemir Kabalci, 2026), or wrote the results without performing any calculations, saying they saw it this way in books. Students experienced more misconceptions in the questions, which require different thinking skills and conceptual knowledge.

Recommendations

This research aims to determine what eighth-graders don’t understand about exponential expressions and why. We shared ideas for correcting these students’ misconceptions. Since mathematics is a cumulative field of study, the learning of each new concept lays the groundwork for the subsequent concepts. Therefore, incomplete and incorrect information formed during the learning of a concept can cause difficulties in learning subsequent concepts and lead to misconceptions. Every middle school mathematics curriculum includes the topic of exponentials. Therefore, we should assess eighth-grade students’ readiness and address any exponent-related misconceptions before they begin learning about exponents. We should administer questions that comfortably allow students to explain and apply their procedural knowledge. Based on their explanations, incorrect statements should be corrected, and more comprehensive questions that measure procedural knowledge and allow them to use their conceptual knowledge should be asked. When conducting written exams for assessment and evaluation, open-ended questions should be preferred so that students’ explanations regarding the questions can be seen in detail; if there are any misconceptions in their explanations, feedback should be provided to the students. Students who encounter the concept of negative exponents for the first time in the eighth-grade have been observed to mistakenly generalize by thinking of the negative exponent as the sign of the base number and solving it as if it were the repeated multiplication of integers. By recalling how the expansion of any integer raised to several integer powers progresses and revisiting how it unfolds—what happens at the zero exponent and what happens when a negative exponent is reached—the student is made aware. This process shows that the sign of the exponential expression remains unaffected by the negative exponent. Since the exponents of integers are taken, they should be guaranteed to solve the problems by paying attention to the concepts of parentheses, absolute value, and whether the exponent is even or odd. If there are any misconceptions on these topics, they are being addressed. When students encounter the situation of raising a power to another integer power (power of a power) and struggle or have misconceptions, they should not be given the rule directly. Again, emphasizing parentheses encourages students to expand the expressions within them, while applying the exponent outside indicates that repeated multiplication equals the product of the exponents. To address the misconceptions observed in the addition and subtraction of exponents, it is important to emphasize the conditions under which students will perform these operations. If necessary, the terms of the operations are broken down one by one to ensure that students understand the rules themselves. Examining similar steps can help students overcome their misconceptions when encountering multiplication and division operations with exponential expressions. The study addressed the misconceptions, their causes, and proposed solutions for eighth-grade students. Another study at different grade levels could address the eighth-grade misconceptions. In addition to the types of questions on the success test for exponential and radical expressions, the research can be expanded by asking students about the other things they know about these two topics and what they don’t understand about them. To make teaching more effective and lasting and to provide better solutions to students’ misconceptions, in-service seminars can be organized separately for high school and middle school groups under the title of misconceptions in mathematics, their causes, and what needs to be done to address them. Instead of conveying rules and shortcuts through a verbal explanation to speed up learning of the subject, students can be centered on the topic from the beginning, allowing them to discover shortcuts and operational properties themselves.

Footnotes

Acknowledgements

The authors are grateful to the middle school students who participated in the data collection, acknowledging their significant contributions to the compilation. This study utilized generative tools, such as Grammarly, to enhance language proficiency. This study is a part of the master’s thesis citation as “Akdemir Kabalci (2025)” prepared by Semanur AKDEMİR KABALCI under the supervision of Sevim SEVGİ.

ORCID iD

Sevim Sevgi

Ethical Considerations

This study received ethical approval from the Research Ethics Committee of Erciyes University (Kayseri, Turkey) on April 25, 2023 (Approval No. 131). This research was conducted ethically. The Ministry of National Education of Turkey approved the study with the number E-47882400-602.04.01-77149951 on May 31st, 2023.

Consent to Participate

Informed consent was obtained verbally and in writing before participation. The consent was saved and included the communication of the study’s purpose, the participants’ right to decline or withdraw, and an assurance of anonymity. All participants (mother and/or father of the students) provided consent for the analysis and publication of their anonymized data in this article.

Author Contributions

First author and second author designed the study, conducted the data collection, piloted the analysis, drafted, and revised the manuscript. Sevim SEVGİ supervised the methodology, discussed preliminary results with Semanur Akdemir KABALCI, provided feedback on the first draft of the manuscript, and made revisions to the article. Both authors approved the final manuscript.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The data used and analyzed during this study are available from the corresponding author upon reasonable request.*

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