f.2 math

Hi ! I am lop****** , feel happy to answer your question.

Q : expend (a-b)(a+b)(a^2+b^2)(a^4+b^4)(a^8+b^8)(a^16+b^16).....(a^1024+b^1024)

A :

By th formula (x+y)(x-y) = x^2 - y^2 and (x^n)^2 = x^2n :

(a-b)(a+b)(a^2+b^2)(a^4+b^4)(a^8+b^8)(a^16+b^16).....(a^1024+b^1024)

= (a^2-b^2)(a^2+b^2)(a^4+b^4)(a^8+b^8)(a^16+b^16)(a^32+b^32)(a^64+b^64)(a^128+b^128)(a^256+b^256)(a^512+b^512)(a^1024+b^1024)

= (a^4-b^4)(a^4+b^4)(a^8+b^8)(a^16+b^16)(a^32+b^32)(a^64+b^64)(a^128+b^128)(a^256+b^256)(a^512+b^512)(a^1024+b^1024)

= (a^8-b^8)(a^8+b^8)(a^16+b^16)(a^32+b^32)(a^64+b^64)(a^128+b^128)(a^256+b^256)(a^512+b^512)(a^1024+b^1024)

= (a^16-b^16)(a^16+b^16)(a^32+b^32)(a^64+b^64)(a^128+b^128)(a^256+b^256)(a^512+b^512)(a^1024+b^1024)

= (a^32-b^32)(a^32+b^32)(a^64+b^64)(a^128+b^128)(a^256+b^256)(a^512+b^512)(a^1024+b^1024)

= (a^64-b^64)(a^64+b^64)(a^128+b^128)(a^256+b^256)(a^512+b^512)(a^1024+b^1024)

= (a^128-b^128)(a^128+b^128)(a^256+b^256)(a^512+b^512)(a^1024+b^1024)

= (a^256-b^256)(a^256+b^256)(a^512+b^512)(a^1024+b^1024)

= (a^512-b^512)(a^512+b^512)(a^1024+b^1024)

= (a^1024-b^1024)(a^1024+b^1024)

= (a^1024)^2 - (b^1024)^2

= a^2048 - b^2048

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Simpler way :

(a-b)(a+b)(a^2+b^2)(a^4+b^4)(a^8+b^8)(a^16+b^16).....(a^1024+b^1024)

Note that (a-b)(a+b) = a^2 - b^2 and we know (a^2-b^2)(a^2+b^2) = (a^2)^2 - (b^2)^2 = a^4 - b^4 , also (a^4-b^4)(a^4+b^4) = (a^4)^2 - (b^4)^2 = a^8 - b^8 ...

Do first two steps :

(a-b)(a+b)(a^2+b^2)(a^4+b^4)(a^8+b^8)(a^16+b^16).....(a^1024+b^1024)

= (a^2-b^2)(a^2+b^2)(a^4+b^4)(a^8+b^8)(a^16+b^16).....(a^1024+b^1024)

= (a^4-b^4)(a^4+b^4)(a^8+b^8)(a^16+b^16).....(a^1024+b^1024)

see the yellow , so we know we can simplify into :

(a^1024)^2 - (b^1024)^2 = a^(2×1024) - b^(2×1024) = a^2048 - b^2048
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To conclude , we get the answer :
(a-b)(a+b)(a^2+b^2)(a^4+b^4)(a^8+b^8)(a^16+b^16).....(a^1024+b^1024) =
a^2048 - b^0248



2011-07-19 20:40:22 補充：
To conclude , we get the answer :
(a-b)(a+b)(a^2+b^2)(a^4+b^4)(a^8+b^8)(a^16+b^16).....(a^1024+b^1024) =
a^2048 - b^2048

timmy的答案應該不對

(a-b)(a+b)=a^2-b^2
(a^2+b^2)(a^2-b^2)=a^4-b^4
.
.
.
so (a-b)(a+b)(a^2+b^2)(a^4+b^4)(a^8+b^8)(a^16+b^16).....(a^1024+b^1024)
=(a^1024-b^1024)(a^1024+b^1024)
=a^2048-b^2048