Question

Find the fifth roots of 32(cos 280° + i sin 280°).

Answer 1

Use the formula: r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ] where k = 0,1,2,3,4

First 5th root: k = 0
r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ]
  (32)^(1/5)*[ cos( (280+360*k)/5 ) + i*sin( (280+360*k)/5 ) ]
  (32)^(1/5)*[ cos( (280+360*0)/5 ) + i*sin( (280+360*0)/5 ) ]
   2*[ cos( (280+360*0)/5 ) + i*sin( (280+360*0)/5 ) ]
  2*[ cos( (280+0)/5 ) + i*sin( (280+0)/5 ) ]
  2*[ cos( 280/5 ) + i*sin( 280/5 ) ]
  2*[ cos( 56 ) + i*sin( 56 ) ]
   -------------------------------------------------------------------
 Second 5th root: k = 1
  r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ]
  (32)^(1/5)*[ cos( (280+360*k)/5 ) + i*sin( (280+360*k)/5 ) ]
  (32)^(1/5)*[ cos( (280+360*1)/5 ) + i*sin( (280+360*1)/5 ) ]
  2*[ cos( (280+360*1)/5 ) + i*sin( (280+360*1)/5 ) ]
  2*[ cos( (280+360)/5 ) + i*sin( (280+360)/5 ) ]
  2*[ cos( 640/5 ) + i*sin( 640/5 ) ]
  2*[ cos( 128 ) + i*sin( 128 ) ]
 -------------------------------------------------------------------
 Third 5th root: k = 2
  r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ]
  (32)^(1/5)*[ cos( (280+360*k)/5 ) + i*sin( (280+360*k)/5 ) ]
  (32)^(1/5)*[ cos( (280+360*2)/5 ) + i*sin( (280+360*2)/5 ) ]
  2*[ cos( (280+360*2)/5 ) + i*sin( (280+360*2)/5 ) ]
   2*[ cos( (280+720)/5 ) + i*sin( (280+720)/5 ) ]
  2*[ cos( 1000/5 ) + i*sin( 1000/5 ) ]
  2*[ cos( 200 ) + i*sin( 200 ) ]
  -------------------------------------------------------------------
 Fourth 5th root: k = 3
  r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ]
  (32)^(1/5)*[ cos( (280+360*k)/5 ) + i*sin( (280+360*k)/5 ) ]
  (32)^(1/5)*[ cos( (280+360*3)/5 ) + i*sin( (280+360*3)/5 ) ]
  2*[ cos( (280+360*3)/5 ) + i*sin( (280+360*3)/5 ) ]
  2*[ cos( (280+1080)/5 ) + i*sin( (280+1080)/5 ) ]
  2*[ cos( 1360/5 ) + i*sin( 1360/5 ) ]
  2*[ cos( 272 ) + i*sin( 272 ) ]
   -------------------------------------------------------------------
 Fifth 5th root: k = 4
  r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ]
  (32)^(1/5)*[ cos( (280+360*k)/5 ) + i*sin( (280+360*k)/5 ) ]
  (32)^(1/5)*[ cos( (280+360*4)/5 ) + i*sin( (280+360*4)/5 ) ]
  2*[ cos( (280+360*4)/5 ) + i*sin( (280+360*4)/5 ) ]
  2*[ cos( (280+1440)/5 ) + i*sin( (280+1440)/5 ) ]
  2*[ cos( 1720/5 ) + i*sin( 1720/5 ) ]
  2*[ cos( 344 ) + i*sin( 344 ) ]