數學疑問~有關化簡!

[匿名]

2010-08-16 3:20 am

1. ( x + 2^1/2 )^5 - ( x - 2^1/2 )^5

2. ( x^1/2 + 3 )^6 + ( x^1/2 - 3 )^6

唔係幾明,唔該幫幫手!
第一題係 [ ( x + 開方2 ) 既五次方 ] - [ ( x - 開方2 ) 既五次方 ]
第二題係 [ ( 開方x + 3 ) 既六次方 ] - [ ( 開方x - 3 ) 既六次方 ]
唔該大家!

回答 (1)

用戶2053

2010-08-16 3:58 am

✔ 最佳答案

1) Let y be 2 ^ 1/2 :(x + y)^5 - (x - y)^5
= (x^5 + 5(x^4)y + 10(x^3)y^2 + 10(x^2)y^3 + 5xy^4 + y^5)
- (x^5 - 5(x^4)y + 10(x^3)y^2 - 10(x^2)y^3 + 5xy^4 - y^5)
= 10(x^4)y + 20(x^2)y^3 + 2y^5= 2y[5x^4 + 10(x^2)y^2 + y^4]= 2(2^0.5) [5x^4 + 10(x^2)(2) + 4]= 2(2^0.5) (5x^4 + 20x^2 + 4)
2) Let y be x^0.5 :( x^1/2 + 3 )^6 + ( x^1/2 - 3 )^6
= (y + 3)^6 + (y - 3)^6
= (y^6 + 6(y^5)3 + 15(y^4)9 + 20(y^3)27 + 15(y^2)81 + 6y(243) + 729)
- (y^6 - 6(y^5)3 + 15(y^4)9 - 20(y^3)27 + 15(y^2)81 - 6y(243) + 729)
= 36y^5 + 1080y^3 + 2916y= 36y(y^4 + 30y^2 + 81)= 36y(y^2 + 3)(y^2 + 27)= 36(x^0.5)(x + 3)(x + 27)

2010-08-15 20:05:26 補充：
2) Let y be x^0.5 :
( x^1/2 + 3 )^6 - ( x^1/2 - 3 )^6

= (y + 3)^6 - (y - 3)^6

=

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