学会逆向运用『幂的运算性质』

幂的运算有四个性质,即同底数幂的乘法性质、幂的乘方性质、积的乘方性质和同底数幂的除法性质.它们是整式乘法的基础和主要依据,四个运算性质反过来也是成立的,在解题时能正反灵活地运用幂的运算性质,会给解题带来很大的帮助. 一、同底数幂的乘法公式的逆向运用 逆用同底数幂的乘法法则,可以把一个幂分解成两个(或两个以上)同底数幂的积.用式子表示为:am+n=am·an(m,n都是正整数).其中,拆分所得的(两个或两个以上)同底数幂的底数与原来幂的底数相同,指数之和等于原来幂的指数.     

有理数的乘法运算策略几例

有理数乘法运算是继加法和减法运算后的又一种运算,也是有理数除法运算和乘方运算的基础,学好有理数乘法运算是学好有理数运算的关键,在进行有理数乘法运算时,要注意根据题目的特点,灵活选取合理的方法,才     

模糊数和模糊值函数的等式限定运算

通过模糊数的结构元表示方法,利用两个单调函数的自反单调变换构造了等式限定算子,推广了文[6]中的等式限定运算,处理了存在负模糊情况下关于乘法运算的不可逆问题.同时,本文还将等式限定运算推广到模糊值函数上,提出了模糊值函数等式限定运算的结构元方法,解决了模糊值函数运算的不可逆问题.     

整体建构,用高质量问题驱动学程——以“幂的运算性质”单元教学为例

幂的运算性质是整式乘法起始阶段的重要内容,由于教材上将同底数幂的运算性质、积的乘方分开编排,所以相关版本的教辅资料上也照此分割课时,造成几种幂的运算性质在教学时较孤立,学生学习幂的运算性质缺少整体观.基于上述理解,我们在最近一次教研活动中,"学材再建构"(著名特级教师李庾南语),从乘方运算出发,引导学生探究归纳出同底数幂的运算性质,再进一步借用乘方的意义生成幂的乘方、积的乘方,取得了较好的教学效果.本文先梳理该课教学活动,并阐释教学立意,供研讨.     

分式混合运算的整数幂形式

<正>同学们在学习分式之前,已经学习过正整数指数幂和零指数幂,同时还学习了5条运算性质,其中对于同底数幂的除法,要求被除式大于除式的指数.在本章引入负整数指数幂以后,整数指数幂的5条运算性质,实际上可以转化为3条.关键是负整数指数幂可以使除法转化为乘法,商转化为积.但是本章对于负整数指数幂的应用仅限于简单的运算,     

中考中幂的运算常见错误分析

<正>由于幂的运算法则多,形式又很类似,如果对幂的运算公式理解不深刻,再加上知识间的相互干扰,在中考解题时,常会出现一些错误,现收集同学们在中考中解此类问题常见的错误剖析如下.一、混淆同底数幂的乘法与幂的乘方致误     

基于交叉影响的IFWGA算子及其在多属性决策中的应用

不同直觉模糊数在信息集结过程中,其隶属度与非隶属度之间可能存在着相互影响.提出了直觉模糊数上的改进的乘法运算和幂运算,重新给出了直觉模糊加权几何平均算子和直觉模糊有序加权几何平均算子的表达式,并研究了他们的一些性质.最后通过实例说明了新的IFWGA集成算子在多属性决策中的应用是可行和有效的.     

模糊数的运算法则

给出模糊数加、减、乘、除运算的较简便的计算方法。     

单源模糊糊及其运算

本文首先指出模糊数的运算存在模糊源问题，然后定义了单源模糊数的一些基本概念，并建立了单源模糊数的运算方法，最后给出了单源模糊数线性方程组的求解方法。     

模糊数的一种新运算方法及其若干应用

首先给出了模糊数的一种新的函数表示定理;基于该表示定理 ,提出了模糊数的一种新运算方法并将其直接应用到模糊数理论中若干问题的讨论,包括:模糊数的差问题,绝对值问题以及模糊数的确界问题 .     

L-Fuzzy Domain及其相关性质

基于[5]提出的L-fuzzy拟序集，引入L-fuzzy集关于L-fuzzy偏序的并，当L是完全分配格时L-fuzzy拟序集上的L-fuzzy定向集等概念，在此基础上定义L-fuzzy domain，证明它是通常Domain的模糊推广，并得到若干相关性质。     

Almost Fuzzy Compactness in L-fuzzy Top ological Spaces

In this paper, the notion of almost fuzzy compactness is defined in L-fuzzy topological spaces by means of inequality, where L is a completely distributive DeMorgan algebra. Its properties are discusse...     

Arithmetical Identities in a 2‐element Model of Tarski's System

All arithmetical identities involving 1, addition, multiplication and exponentiation will be true in a 2‐element model of Tarski's system if a certain sequence of natural numbers is not bounded. That sequence can be bounded only if the set of Fermat's prime numbers is finite.     

Similarity of L-fuzzy Relations Based on L-topologies Induced by L-fuzzy Rough Approximation Operators

This paper proposes similarity of L-fuzzy relations based on L-topologies induced by L-fuzzy rough approximation operators. First, the notion L-fuzzy rough set is generalized and the relationship between generalized L-fuzzy rough sets and L-topologies on an arbitrary universe is investigated. It shows that Alexandrov L-topologies can be induced by L-fuzzy relations without any preconditions. Second, the concept of similarity of L-fuzzy relations is introduced and variations of an L-fuzzy relation are investigated. Third, algebraic structures on similarity of L-fuzzy relations are obtained. Finally, we prove that the subset of the transitive L-fuzzy relations similar to a fixed L-fuzzy relation is a complete distributive lattice.     

Intermediate arithmetic operations on ordinal numbers

There are two well‐known ways of doing arithmetic with ordinal numbers: the “ordinary” addition, multiplication, and exponentiation, which are defined by transfinite iteration; and the “natural” (or “Hessenberg”) addition and multiplication (denoted ⊕ and ⊗), each satisfying its own set of algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of multiplying ordinals (denoted × ), defined by transfinite iteration of natural addition, as well as the notion of exponentiation defined by transfinite iteration of his multiplication, which we denote . (Jacobsthal's multiplication was later rediscovered by Conway.) Jacobsthal showed these operations too obeyed algebraic laws. In this paper, we pick up where Jacobsthal left off by considering the notion of exponentiation obtained by transfinitely iterating natural multiplication instead; we shall denote this . We show that and that ; note the use of Jacobsthal's multiplication in the latter. We also demonstrate the impossibility of defining a “natural exponentiation” satisfying reasonable algebraic laws.     

A solution to identities problem in 2‐element HSI‐algebras

All arithmetical identities involving 1, addition, multiplication and exponentiation are valid in every 2‐element HSI‐algebra. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zeroless arithmetic: representing integers ONLY using ONE

We use recurrence equations (alias difference equations) to enumerate the number of formula representations of positive integers using only addition and multiplication, and using addition, multiplication and exponentiation, where all the inputs are ones. We also describe efficient algorithms for the random generation of such representations, and use dynamical programming to find a shortest possible formula representing any given positive integer.     

Montgomery Multiplication in GF(2k)

We show that the multiplication operation c=a · b · r^-1 in the field GF(2^k can be implemented significantly faster in software than the standard multiplication, where r is a special fixed element of the field. This operation is the finite field analogue of the Montgomery multiplication for modular multiplication of integers. We give the bit-level and word-level algorithms for computing the product, perform a thorough performance analysis, and compare the algorithm to the standard multiplication algorithm in GF(2^k. The Montgomery multiplication can be used to obtain fast software implementations of the discrete exponentiation operation, and is particularly suitable for cryptographic applications where k is large.     

Ordinal Operations on Surreal Numbers

An open problem posed by John H. Conway in [2] was whether onecould, on his system of numbers and games, ‘... defineoperations of addition and multiplication which will restricton the ordinals to give their usual operations’. Sucha definition for addition was later given in [4], and this paperwill show that a definition also exists for multiplication.An ordinal exponentiation operation is also considered.     

Questions of cardinality of finite fuzzy sets

In this paper we focus our attention on finite fuzzy sets. A complete, simple and easily applicable cardinality theory for them is presented. Questions of equipotency and non-classically understood cardinal numbers of finite fuzzy sets are discussed in detail. Also, problems of arithmetical operations (addition, subtraction, multiplication, division, and exponentiation) on as well as ordering relation for those cardinals are carefully investigated.