Question

Find the solution of the following equation whose argument is strictly between 270^\circ270 ∘ 270, degree and 360^\circ360 ∘ 360, degree. Round your answer to the nearest thousandth. z^4=-625z 4 =−625

Answer 1

`\rightarrow z^4=-625\\\\\rightarrow z=(-625+0i)^{(1)/(4)}\\\\\rightarrow x+iy=(-625+0i)^{(1)/(4)}\\\\ x=r \cos A\\\\y=r \sin A\\\\r \cos A=-625\\\\ r \sin A=0\\\\x^2+y^2=625^(2)\\\\r^2=625^(2)\\\\|r|=625\\\\ \tan A=(0)/(-625)\\\\ \tan A=0\\\\ A=\pi\\\\\rightarrow z= [625(\cos (2k \pi+pi) +i \sin (2k\pi+ \pi)]^{(1)/(4)}\\\\k=0,1,2,3,4,....\\\\\rightarrow z=(625)^{(1)/(4)}[\cos ((2k \pi+pi))/(4) +i \sin ((2k\pi+ \pi))/(4)]`

`\rightarrow z_(0)=(625)^{(1)/(4)}[\cos (pi)/(4) +i \sin (\pi))/(4)]\\\\\rightarrow z_(1)=(625)^{(1)/(4)}[\cos (3\pi)/(4) +i \sin (3\pi)/(4)]\\\\ \rightarrow z_(2)=(625)^{(1)/(4)}[\cos (5\pi)/(4) +i \sin (5\pi)/(4)]\\\\ \rightarrow z_(3)=(625)^{(1)/(4)}[\cos (7\pi)/(4) +i \sin (7\pi)/(4)]`

Argument of Complex number

Z=x+iy , is given by

If, x>0, y>0, Angle lies in first Quadrant.

If, x<0, y>0, Angle lies in Second Quadrant.

If, x<0, y<0, Angle lies in third Quadrant.

If, x>0, y<0, Angle lies in fourth Quadrant.

We have to find those roots among four roots whose argument is between 270° and 360°.So, that root is

`\rightarrow z_(2)=(625)^{(1)/(4)}[\cos (5\pi)/(4) +i \sin (5\pi)/(4)]`