MQ44 --- Denest－飛航模式

標題:

MQ44 --- Denest

發問:

MQ44 --- DenestDifficulty: 50%Denest √(?5 - ?4).

最佳解答:

?5 - ?4 = (5 - 4) / (?52 + ?5 ?4 + ?42) ) ......... ∵ a - b = (a3 - b3) / (a2 + ab + b2) = 1 / (?25 + ?20 + 2?2)Note that 25 * 20 * 2 = 1000 is a perfect cube. To rationalize the denominator, assuming that (?25 + ?20 + 2?2) (a?25 + b?20 + c?2)2 is a rational number while the numerator is a perfect square = (a?25 + b?20 + c?2)2 . (?25 + ?20 + 2?2) (a?25 + b?20 + c?2)2= (?25 + ?20 + 2?2) (a2?625 + b2?400 + c2?4 + 2ab?500 + 2bc?40 + 2ca?50)= (?25 + ?20 + 2?2) (5a2?5 + 2b2?50 + c2?4 + 10ab?4 + 4bc?5 + 2ca?50)= (?25 + ?20 + 2?2) ( (5a2+4bc)?5 + (2b2+2ca)?50 + (c2+10ab)?4 )= (5a2+4bc)?125 + (2b2+2ca)?1250 + (c2+10ab)?100 + (5a2+4bc)?100 + (2b2+2ca)?1000 + (c2+10ab)?80 + 2(5a2+4bc)?10 + 2(2b2+2ca)?100 + 2(c2+10ab)?8= 5(5a2+4bc) + 5(2b2+2ca)?10 + (c2+10ab)?100 + (5a2+4bc)?100 + 10(2b2+2ca) + 2(c2+10ab)?10 + 2(5a2+4bc)?10 + 2(2b2+2ca)?100 + 4(c2+10ab)= 5(5a2+4bc) + 10(2b2+2ca) + 4(c2+10ab) + ( 5(2b2+2ca) + 2(c2+10ab) + 2(5a2+4bc) )?10 + ( (c2+10ab) + (5a2+4bc) + 2(2b2+2ca) )?100For it is rational , ( 5(2b2+2ca) + 2(c2+10ab) + 2(5a2+4bc) )?10 = 0 { ( (c2+10ab) + (5a2+4bc) + 2(2b2+2ca) )?100 = 0 ? 5(b2+ca) + (c2+10ab) + (5a2+4bc) = 0 ... (1) { (c2+10ab) + (5a2+4bc) + 2(2b2+2ca) = 0 ... (2)(1) - (2) : 5(b2+ca) - 2(2b2+2ca) = 0 b2 = - ac Substitute into (1) or (2) : (c2+10ab) + (5a2+4bc) = 0 ... (3)Let a = - bk , c = b/k , substitute into (3) :(b2/k2 - 10b2k) + (5b2k2 + 4b2/k) = 0 b2 (1/k2 - 10k + 5k2 + 4/k) = 0 1/k2 - 10k + 5k2 + 4/k = 0 since b2 ≠ 0 otherwise a = b = c = 0 , lead the denominator = 0.5k?- 10k3 + 4k + 1 = 0 (k - 1) (5k3 - 5k2 - 5k - 1) = 0 k = 1 or k = 1.67035674... (irrational , ignore it)Put k = 1 , then a = - b , c = b , i.e. - a = b = cSet - a = b = c = 1 , the denominator = 5(5a2+4bc) + 10(2b2+2ca) + 4(c2+10ab) = 5(5 + 4) + 10(2 - 2) + 4(1 - 10) = 9∴ 1 / (?25 + ?20 + 2?2) = (- ?25 + ?20 + ?2)2 / 9 then √(?5 - ?4) = 1 / √(?25 + ?20 + 2?2) = (- ?25 + ?20 + ?2) / 3

其他解答:

Denest a nest radical root means to simplify a nest radical root.|||||what is denest?