• Communications of Mathematical Education
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• v.25 no.2
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• pp.323-339
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• 2011

The expansion of exponential law as the law of calculation of integer numbers can be a good material for the students to experience an extended configuration which is based on an algebraic principle of the performance of equivalent forms. While current textbooks described that exponential law can be expanded from natural number to integer, rational number and real number, most teachers force students to accept intuitively that the exponential law is valid although exponent is expanded into real number. However most teachers overlook explaining the value of exponent of rational number or exponent of irrational number so most students have a lot of questions whether this value is a rational number or a irrational number. Related to students' questions, most teacher said that it is out of the current curriculum and students will learn it after going to college instead of detailed answers. In this paper, we will present several examples and the values about irrational exponents of a positive rational and irrational exponents of a positive irrational number, and study the recognition and fallacy of would-be teachers about the cases of irrational exponents of a positive rational and irrational exponents of a positive irrational number at the expansion of exponential law.

• Journal of the Korean School Mathematics Society
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• v.16 no.3
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• pp.583-600
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• 2013

In this paper, we study the recognition and fallacy of would-be in-service teachers about numbers with irrational exponent. We chose 51 secondary school teachers who are teaching mathematics in K metropolitan city and investigate their recognition and fallacy about the cases of irrational exponents of a positive rational and irrational exponents of a positive irrational number at the expansion of exponential law. We found following facts. First, in-service teacher's a percentage of correct answers differ depending on the type of numbers with irrational exponent. Second, in-service teachers decide their answer depending on intuition rather than logic. Third, in-service teachers decide their answer depending on exponential rather than base.

• Steel and Composite Structures
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• v.18 no.4
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• pp.909-924
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• 2015

This paper investigates the vibration phenomenon of a nanobeam subjected to a time-dependent heat flux. Material properties of the nanobeam are assumed to be graded in the thickness direction according to a novel exponential distribution law in terms of the volume fractions of the metal and ceramic constituents. The upper surface of the functionally graded (FG) nanobeam is pure ceramic whereas the lower surface is pure metal. A nonlocal generalized thermoelasticity theory with dual-phase-lag (DPL) model is used to solve this problem. The theories of coupled thermoelasticity, generalized thermoelasticity with one relaxation time, and without energy dissipation can extracted as limited and special cases of the present model. An analytical technique based on Laplace transform is used to calculate the variation of deflection and temperature. The inverse of Laplace transforms are computed numerically using Fourier expansion techniques. The effects of the phase-lags (PLs), nonlocal parameter and the angular frequency of oscillation of the heat flux on the lateral vibration, the temperature, and the axial displacement of the nanobeam are studied.

• Structural Engineering and Mechanics
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• v.51 no.2
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• pp.199-214
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• 2014

The present investigation is concerned with the effect of two temperatures on functionally graded (FG) nanobeams subjected to sinusoidal pulse heating sources. Material properties of the nanobeam are assumed to be graded in the thickness direction according to a novel exponential distribution law in terms of the volume fractions of the metal and ceramic constituents. The upper surface of the FG nanobeam is fully ceramic whereas the lower surface is fully metal. The generalized two-temperature nonlocal theory of thermoelasticity in the context of Lord and Shulman's (LS) model is used to solve this problem. The governing equations are solved in the Laplace transformation domain. The inversion of the Laplace transformation is computed numerically using a method based on Fourier series expansion technique. Some comparisons have been shown to estimate the effects of the nonlocal parameter, the temperature discrepancy and the pulse width of the sinusoidal pulse. Additional results across the thickness of the nanobeam are presented graphically.