分类：教育/科学-理工学科-数学
问题：已知w是1的七次方根。求w/(1+w^2)+w^2/(1+w^4? 
已知w是1的七次方根。求w/(1+w^2)+w^2/(1+w^4)+w^3/(1+w^6)的值。请详解
回答：题：已知w是1的七次方根。求w/(1+w^2)+w^2/(1+w^4)+w^3/(1+w^6)的值。请详解
解：
已知w是1的七次方根，则有两种可能：
其一，w=1，则w/(1+w^2)+w^2/(1+w^4)+w^3/(1+w^6)=3/2．
其二，w为非1的复根，则满足下列条件
1+w+w^2+w^3+w^4+w^5+w^6=0，w^7=1．
根据此条件进行简化，则有
w/(1+w^2)+w^2/(1+w^4)+w^3/(1+w^6)
=(w(1+w^4)(1+w^6)+w^2(1+w^2)(1+w^6)+w^3(1+w^2)(1+w^4))/((1+w^2)(1+w^4)(1+w^6))
=2(1+w+w^2+w^3+w^4+w^5)/(1+w+w^2+w^3+w^4+w^5+2w^6)
= -2w^6/w^6
= -2.
解毕．