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分母有理化是化简二次根式的常规方法.但对于一些题目,若转换一种思维方式,运用发散思维,另辟解题途径,不用分母有理化,其过程反而简捷明快.下面举例说明.一、巧妙约简,耳目一新例1 化简:6-6~(1/6)/6~(1/6)-1+7-7~(1/7)/7~(1/7)-1+10-10~(1/10)/10~(1/10)-1.分析 因为6-6~(1/6)=6~(1/6)(6~(1/6)-1),7-7~(1/7)=7~(1/7)(7~(1/7)-1),10-10~(1/10)=10~(1/10)(10~(1/10)-1),从而可巧妙约简.解 原式=6~(1/6)(6~(1/6)-1)/6~(1/6)-1+7~(1/7)(7~(1/7)-1)/7~(1/7)-1+10~(1/10)(10~(1/10)-1)/10~(1/10)-1=6~(1/6)+7~(1/7)+10~(1/10).二、巧妙通分,简捷明快
The rationalization of the denominator is a conventional method of simplifying the secondary roots. However, for some problems, if we change one way of thinking, use divergent thinking, open up another way to solve the problem, and do not need rationalization of the denominator, the process is concise. The following examples illustrate. First, clever reduction, a new example of a simplification: 6-6~(1/6)/6~(1/6)-1+7-7~(1/7)/7~(1/7)- 1+10-10~(1/10)/10~(1/10)-1. Analysis because 6-6~(1/6)=6~(1/6)(6~(1/6)- 1),7-7~(1/7)=7~(1/7)(7~(1/7)-1),10-10~(1/10)=10~(1/10)( 10~(1/10)-1), which can be used to reduce the complexity of the solution. Original solution = 6~(1/6)(6~(1/6)-1)/6~(1/6)-1+ 7~(1/7)(7~(1/7)-1)/7~(1/7)-1+10~(1/10)(10~(1/10)-1)/10~ (1/10)-1=6~(1/6)+7~(1/7)+10~(1/10). Second, sub-passage, simple and clear