设ω=（-1+√3i）/2,试求下列各值（1）ω^2(2)ω^33.(1+ω+ω^2)4.1/ω5.(ω+2ω^2)^2+(2ω+ω^2)^26.(5-3ω-3ω^2)(7-2ω+7ω)^2(4+4ω-ω^2)

Question

设ω=（-1+√3 i）/2 ,试求下列各值 （1）ω^2 (2)ω^3 3.(1+ω+ω^2) 4.1/ω
5.(ω+2ω^2)^2+(2ω+ω^2)^2 6.(5-3ω-3ω^2)(7-2ω+7ω)^2(4+4ω-ω^2)

Answer 1

ω=(-1+√3i)/2
ω²=[(-1+√3i)/2]²=(1-3-2√3i)/4=(-1-√3i)/2
ω³=ω*ω²=(-1+√3i)/2*(-1-√3i)/2=1
1+ω+ω²=1+(-1+√3i)/2+(-1-√3i)/2=0
1/ω=ω²/ω³=(-1-√3i)/2
(ω+2ω²)²+(2ω+ω²)²
=ω+4ω^4+4ω³+4ω²+ω^4+4ω³
=ω+4ω+4+4ω²+ω+4
=4ω²+6ω+8
=4(1+ω+ω²)+2ω+4
=2ω+4
=3+√3i
(5-3ω-3ω²)(7-2ω+7ω)²(4+4ω-ω²)
=(8-3-3ω-3ω²)(7+7ω+7ω²-9ω)(4+4ω+4ω²-5ω²)
=[8-3(1+ω+ω²)][7(1+ω+ω²)-9ω][4(1+ω+ω²)-5ω²]
=8(-9ω)(-5ω²)
=360