Question

(a+1/a)(a^2+1/a^2)(a^4+1/a^4)(a^8+1/a^8)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)

Answer 1

乘以(a-1/a)，再除以(a-1/a)，利用平方差公式
=(a-1/a)(a+1/a)(a^2+1/a^2)(a^4+1/a^4)(a^8+1/a^8)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)/(a-1/a)
=(a^2-1/a^2)(a^2+1/a^2)(a^4+1/a^4)(a^8+1/a^8)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)/(a-1/a)
=(a^4-1/a^4)(a^4+1/a^4)(a^8+1/a^8)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)/(a-1/a)
=(a^8-1/a^8)(a^8+1/a^8)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)/(a-1/a)
=(a^16-1/a^16)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)/(a-1/a)
=(a^32-1/a^32)(a^32+1/a^32)(a^2-1)/(a-1/a)
=(a^64-1/a^64)(a^2-1)/(a-1/a)
=a(a^64-1/a^64)(a^2-1)/(a^2-1)
=a(a^64-1/a^64)
=a^65-1/a^63.

Answer 2

原式 = [原式 *（a-1/a）]/（a-1/a） 注：方差公式

=[(a^2-1/a^2)(a^2+1/a^2)(a^4+1/a^4)(a^8+1/a^8)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)]/(a-1/a)
=[(a^4-1/a^4)(a^4+1/a^4)(a^8+1/a^8)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)]/(a-1/a)
=[(a^8-1/a^8)(a^8+1/a^8)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)]/(a-1/a)
=[(a^16-1/a^16)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)]/(a-1/a)
=[(a^32-1/a^32)(a^32+1/a^32)(a^2-1)]/(a-1/a)
=[(a^64-1/a^64)(a^2-1)]/(a-1/a)
分子分母同时乘以a/[a^2-1]
原式=(a^64-1/a^64)*a
=a^65-1/a^63

Answer 3

先乘(a-1/a),再除以(a-1/a)
原式=(a-1/a)(a+1/a)(a²+1/a²)...(a^32+1/a^32)(a²-1)÷(a-1/a)
=(a²-1/a²)(a²+1/a²).....(a^32+1/a^32)(a²-1)÷(a²-1)/a
=......
=(a^32-1/(a^32)(a^32+1/a^32)*a
=(a^64-1/a^64)*a
=a^65-1/a^63

Answer 4

原式=(a-1/a)(a+1/a)(a^2+1/a^2)(a^4+1/a^4)(a^8+1/a^8)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)/(a-1/a)

=(a^2-1/a^2)(a^2+1/a^2)(a^4+1/a^4)(a^8+1/a^8)(a^16+1/a^16)(a^32+1/a^32)(a^2-1)/[(a^2-1)/a]
反复用平方差

=(a^64-1/a^64)(a^2-1)/[(a^2-1)/a]

=a(a^64-1/a^64)
=a^65-1/a^63