數學知識交流 － 求值 (1)

AGREEEEEEE

2012-05-19 14:00:44 補充：
Wait...
I'm now crazying in it

2012-05-19 14:45:52 補充：
THX!!!!!

XD

2012-05-19 17:28:02 補充：
x - y = 1 ---(i)x³ - y³ = 4 ---(ii) ∵ A) Find xy.From (i).x - y = 1x² - 2xy+ y² = 1x² + y² = 1 + 2xy ---(iii)From (ii).x³ - y³ = 4(x - y)(x² + xy+ y²) = 41(x² + xy+ y²) = 4x² + y² = 4 - xy ---(iv)Combine (iii) & (iv).1 + 2xy = 4 - xyxy = 1 ---(v) B) Find x² + y².Sub. (v) into (iii)x² + y² = 1 + 2(1)x² + y² = 3 ---(vi) C) Find x + y.From (vi).x² + y² = 3x² + 2xy+ y² = 3 +2(1) <-[From (v)](x + y)² = 5x + y = ±√5 ---(vii) D) Find x² - y².x² - y²= (x - y)(x + y)= (1)(±√5) <-[From (i) & (vii)]= ±√5 ---(viii) E) Find x³ + y³.x³ + y³= (x + y)(x² - xy+ y²)= ±√5(3 -1) <-[From (v), (vi) & (vii)]= ±2√5 ---(ix) F) Find x⁴ - y⁴.x⁴ - y⁴= (x² - y²)(x² + y²)= ±√5(3) <-[From (vi) & (viii)]= ±3√5 ---(x) G) Find x⁴ + y⁴.From (viii).x² - y² = ±√5x⁴ - 2x²y² + y⁴ = 5x⁴ + y⁴ = 5 + 2x²y² ---(vi)x⁴ + y⁴ = 5 + 2(1)² <-[From (v)]x⁴ + y⁴ = 7 ---(xi) H) Find x⁵ - y⁵.x⁵ - y⁵= (x - y)(x⁴ + x³y + x²y² + xy³ + y⁴)= (x - y)[(x⁴ + y⁴) +xy(x² + y²) + x²y²]= (1)[7 + 1(3) + 1²] <-[From (i), (v), (vi) & (xi)]= 11 ---(xii) I) Find x⁶ - y⁶.x⁶ - y⁶= (x³ - y³)(x³ + y³) <-[From (ii) & (ix)]= 4(±2√5)= ±8√5 ---(xiii) J) Find x⁶ + y⁶.x⁶ + y⁶= (x² + y²)(x⁴ - x²y² +y⁴)= (3)(7 - 1²) <-[From (v), (vi) & (xi)]= 18 ---(xiv) K) Find x⁷ - y⁷.x⁷ - y⁷= (x - y)(x⁶ + x⁵y + x⁴y² + x³y³ + x²y⁴ + xy⁵ + y⁶)= (x - y)[(x⁶ + y⁶) + xy(x⁴ + y⁴) + x²y²(x² + y²) + x³y³)= (1)[18 + 1(7) + 1²(3) + 1³)= 29 ---(xv) L) Find x⁸ + y⁸.From (x).x⁴ - y⁴ = ±3√5x⁸ - 2x⁴y⁴ + y⁸ = 45x⁸ + y⁸ = 45 + 2x⁴y⁴x⁸ + y⁸ = 45 + 2(1)⁴ <-[From (v)]x⁸ + y⁸ = 47 ---(xvi) M) Find x¹º + y¹º.From (xii).x⁵ - y⁵ = 11x¹º - 2x⁵y⁵ + y¹º = 121x¹º + y¹º = 121 + 2x⁵y⁵x¹º + y¹º = 121 + 2(1)⁵ <-[From (v)]x¹º + y¹º = 123 ---(xvii)

2012-05-19 17:29:01 補充：
N) Find x¹² + y¹².
From (xiii).
x⁶ - y⁶ = ±8√5

2012-05-19 17:29:58 補充：
N) Find x¹² + y¹².
From (xiii).
x⁶ - y⁶ = ±8√5
x¹² - 2x⁶y⁶ + y¹² = 320
x¹² + y¹² = 320 + 2x⁶y⁶
x¹² + y¹² = 320 + 2(1)⁶ <-[From (v)]
x¹² + y¹² = 322 ---(xviii)

2012-05-19 17:31:18 補充：
O) Find x¹³ + y¹³.
x¹³ + y¹³
= (x - y)(x¹² + x¹¹y + x¹ºy² + x⁹y³ + x⁸y⁴ + x⁷ y⁵ + x⁶y⁶
+ x⁵y⁷ + x⁴y⁸ + x³y⁹ + x²y¹º + xy¹¹ + y¹²)
= (x - y)[(x¹² + y¹²) + xy(x¹º + y¹º) + x²y²(x⁸ + y⁸)
= (x - y)[x³y³(x⁶ + y⁶) + x⁴y⁴(x⁴ + y⁴) + x⁵y⁵ (x² + y²) + x⁶y⁶]

2012-05-19 17:31:49 補充：
= (1)[322 + 1(123) + 1²(47) + 1³(18) + 1⁴(7) + 1⁴(3) + 1]
= <-[From (v), (vi), (ix), (xiv), (xvi), (xvii) & (xviii)]
= 521 ---(xix)

2012-05-19 17:32:16 補充：
∴

1) x² + y² = 3 <-[From (vi)]
2) x⁴ - y⁴ = ±3√5 <-[From (x)]
3) x⁶ - y⁶ = ±8√5 <-[From (xiii)]
4) x⁷ - y⁷ = 29 <-[From (xv)]
5) x¹³ - y¹³ = 521 <-[From (xix)]

FINALLY FINISH!~

2012-05-19 17:34:18 補充：
You can omit the supplement of my answer.

*** Stupid YAHOO server make me not be able to type more than 2000 words!

2012-05-19 17:42:20 補充：
CORRECTION!

WRONG:
= (x - y)[(x¹² + y¹²) + xy(x¹º + y¹º) + x²y²(x⁸ + y⁸)
= (x - y)[x³y³(x⁶ + y⁶) + x⁴y⁴(x⁴ + y⁴) + x⁵y⁵ (x² + y²) + x⁶y⁶]

CORRECT:
= (x - y)[(x¹² + y¹²) + xy(x¹º + y¹º) + x²y²(x⁸ + y⁸)
x³y³(x⁶ + y⁶) + x⁴y⁴(x⁴ + y⁴) + x⁵y⁵ (x² + y²) + x⁶y⁶]

2012-05-19 21:58:38 補充：
OH YES!!!!!

2012-05-20 19:59:23 補充：
Why it is not clear?

2012-05-21 07:08:20 補充：
-.-
............

All the answer of 自由自在 are correct

2012-05-20 20:15:11 補充：
the comment spreaded, let me can't follow your answer.

For your verification:
3, ±3√5,±8√5,29,521

2012-05-19 21:23:14 補充：
Can be shorter, e.g.
(x^7-y^7 )(x^6+y^6 )=x^13-y^13+(x y)^6 (x-y)
=>(29 )(18)=y^13-y^13+1
=>x^13-y^13=521

So crazy~~

2012-05-19 14:07:13 補充：
ADD OIL ~~~!!