SOLUTION: please help to simplify 3i - 5 DIVIDED BY -i^3 + 3i^2

please help to simplify

  3i - 5
———————————
 -i³ + 3i²

i² = -1 and i³ = i²·i = (-1)·i = -i

So replace i³ by -i and i² by -1

    3i - 5
———————————————
 -(-i) + 3(-1)

  3i - 5
 ————————
   i - 3

Write numerator and denominator
in standard A + Bi order:


  -5 + 3i
 —————————
  -3 + i

Multiply top and bottom by the conjugate
of the denominator, which is -3 - i

  (-5 + 3i)(-3 - i)
 ——————————————————— 
  (-3 + i)(-3 - i)

FOIL out top and bottom:

  15 + 5i - 9i - 3i²
 ————————————————————
   9 + 3i - 3i - i²

Combine terms:


  15 - 4i - 3i²
 ———————————————               
      9 - i²

Replace i² by -1

  15 - 4i - 3(-1)
 —————————————————
      9 - (-1)

  15 - 4i + 3
 —————————————
     9 + 1

 18 - 4i
—————————
   10

Make two fractions:

 18     4i 
———— - ————
 10     10

Reduce the fractions

 9     2i 
——— - ————
 5     5

Write in A + Bi form:

 9     2 
——— - ——— i
 5     5

Edwin