1976年2月1日

【數學】關於最簡根式 ~help~緊急(根式化簡)

【來源】http://tw.knowledge.yahoo.com/question/question?qid=1607082704151
【問題】發問者: ( 初學者 5 級)
以下有3題請幫忙解答THANKS!

第一題:1/(3√16-3√8+3√4)
第2題:1/(3√3-1)
第3題:1/(3√9+3√6+3√4)

P.S.因為不知如何打出根號所,以用中文表示,請見諒!

【我的回答】
這幾題的關鍵是
 立方差公式: a^3 - b^3 = (a-b)(a^2 + ab + b^2)
 立方和公式: a^3 + b^3 = (a+b)(a^2 - ab + b^2)

我們把根號用指數形式、也就是用分數次方項代表,

1. 還沒想到,有時間再補充
2.
1 / [3^(1/3) - 1]
 = [3^(2/3) + 3^(1/3) +1 ] / [3^(1/3) - 1]*[3^(2/3) + 3^(1/3) +1 ]
 = [3^(2/3) + 3^(1/3) +1 ] / [3 - 1 ]
 = [3^(2/3) + 3^(1/3) +1 ] / 2

3.
令 x = 3^(1/3), x^2 = 3^(2/3), x^3 = 3
令 y = 2^(1/3), y^2 = 2^(2/3), y^3 = 2及 x*y = 6^(1/3)
原式 = 1/[9^(1/3) + 6^(1/3) + 4^(1/3)]
   = 1/[3^(2/3) + 6^(1/3) + 2^(2/3)]
   = 1/[x^2 + xy + y^2]
   = (x-y) / (x-y)[x^2 + xy + y^2]
   = (x-y) / (x^3 -y^3)
   = [3^(1/3) - 2^(1/3)] / (3-2)
   = [3^(1/3) - 2^(1/3)]

參考資料 自己

2007-08-27 13:34:57 補充
恩~吃飽飯果然是有 feel~

令 x = 2^(1/3),  x^2 = 2^(2/3) = 4^(1/3),  x^3 = 2 = 8^(1/3),  x^4 = 16^(1/3)
1.
原式
 = 1/ (x^4 - x^3 + x^2)
 = 1/ (x^2)*(x^2 - x^1 + 1)
 = (x+1) / (x^2)*[(x+1)(x^2 - x^1 + 1)]
 = (x+1) / (x^2)*[x^3 + 1]
 = (x+1) / (x^2)*3
 = (1/3)* [ x^(-1) + x^(-2)]
 = (1/3)* [ (1/2)^(1/3) + (1/4)^(1/3)]

參考資料 自己

0 意見:

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