Question

已知:a=4^(1/3)+2^(1/3)+1^(1/3),那么3/a+3/a^2+1/a^3=_____

Answer 1

先化简a利用立方差公式
a=[2^(1/3)]^2+2^(1/3)+1={[2^(1/3)]^3-1}/[2^(1/3)-1]=1/[2^(1/3)-1]
1/a=[2^(1/3)-1], 1+1/a=2^(1/3)
所以，待求式-1+1结果不变
-1+1+(3/a+3/a^2+1/a^3)
= -1+[1^3+3·1^2·(1/a)+3·1·(1/a)^2+(1/a)^3]
= -1+(1+1/a)^3
= -1+[2^(1/3)]^3
= 1

Answer 2

a=4^(1/3)+2^(1/3)+1^(1/3),
[2^(1/3)-1]a=[2^(1/3)-1]*[4^(1/3)+2^(1/3)+1^(1/3)]=2-1=1,
1/a=2^(1/3)-1,
1+1/a=2^(1/3),
1+3/a+2/a^2+1/a^3=2,
3/a+2/a^2+1/a^3=1.