Solution.
1) z4+4=0
z4=−4
eiπ=−1 and let z=reiθ
(reiθ)4=4eiπ
r4ei4θ=4eiπ
sor4=4⟹r=2
4θ=π+2nπ⟹θ=4π+42nπ where n=0,1,2,3
So, roots of the equation will be,
z=2ei(4π),2ei(4π+2π),2ei(4π+π),2ei(4π+23π)
z=2ei(4π),2ei(43π),2ei(45π),2ei(47π)
Since, eiθ=cosθ+isinθ
z1=2ei(4π)=2(cos4π+isin4π)=2(21+i21)=1+i
z2=2ei(43π)=2(cos43π+isin43π)=2(−21+i21)=−1+i
z3=2ei(45π)=2(cos45π+isin45π)=2(−21−i21)=−1−i
z4=2ei(47π)=2(cos47π+isin47π)=2(21−i21)=1−i
z4−4=0⟹(z2−2)(z2+2)=0
z2−2=0⟹z=±2
z2+2=0⟹z2=−2⟹z=±2i
z=2,−2,2i,−2i
2) z8−16=0⟹(z4−4)(z4+4)=0
z=−2,2,−2i,2i,1+i,−1+i,−1−i,1−i
z8+16=0
z8=−16
(reiθ)8=(16eiπ)
Solving it, r=(16)1/8⟹r=2
8θ=π+2nπ⟹θ=8π+82nπ where =0,1,2,3,4,5,6,7
z1=2ei8π=2(cos8π+isin8π)
z2=2ei83π=2(cos83π+isin83π)
z3=2ei85π=2(cos85π+isin85π)
z4=2ei87π=2(cos87π+isin87π)
z5=2ei89π=2(cos89π+isin89π)
z6=2ei811π=2(cos811π+isin811π)
z7=2ei813π=2(cos813π+isin813π)
z8=2ei815π=2(cos815π+isin815π)
Answer:
1)z4+4=0:
z1=1+i;
z2=−1+i;
z3=−1−i;
z4=1−i;
z4−4=0:
z=2,−2,2i,−2iz=2,−2,2i,−2i ;
2)z8−16=0:
z=−2,2,−2i,2i,1+i,−1+i,−1−i,1−i ;
z8+16=0:
z1=2(cos8π+isin8π);
z2=2(cos83π+isin83π);
z3=2(cos85π+isin85π);
z4=2(cos87π+isin87π);
z5=2(cos89π+isin89π);
z6=2(cos811π+isin811π);
z7=2(cos813π+isin813π);
z8=2(cos815π+isin815π).
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