Question

Solvetheequationsbelowandhence,showthesolutionsoftheseequationsontheArganddiagram:(i)z4−i=0(ii)z3=−√3−i(iii)z5=−18i(iv)z=16i1+i18(v)z=1+i√3+i16

Answer 1

(i)$z4-i=0$

$4(x+iy)-i=0\Rightarrow 4x+(4y-1)i=0\Rightarrow x=0 \text{ and }y=\dfrac{1}{4}$

(ii) $z3=-\sqrt{3}-i$

$3(x+iy)+\sqrt{3}+i=0\Rightarrow (3x+\sqrt{3})+i(y+1)=0\Rightarrow x=\dfrac{-1}{\sqrt{3}}, y=-1$

(iii)$z5=-18i$

$5(x+iy)+18i=0\Rightarrow 5x+i(y+18)=0\Rightarrow x=0,y=-18$

(iv) $z=16i+18i$

$x+iy=34i\Rightarrow x=0,y=34$

(v) $z=1+i\sqrt{3}+16i$

$x+iy=1+i(\sqrt{3}+16)\Rightarrow x=1,y=\sqrt{3}+16$