Hi, can you show me how to solve a problem like this one?
  Find Multiplicative Inverse.

      -i
     -----
      3+5i


Is the answer just 3-5i     ????
                  ------
                    i         

No.  You do invert it, but you don't change any signs yet.

It's simply the original problem inverted without changing any
signe

 
3 + 5i
------
  -i 

However that's not finished.

You write the denominator as 0 - 1i

3 + 5i
------
0 - 1i

Then you form the conjugate of the denominator, 0 - 1i.
(This is the only time where you change the sign of the second term.)
The conjugate of 0 - 1i is 0 + 1i.  So you form the fraction by putting
that over itself, so that it just equals 1

0 + 1i
------
0 + 1i

Then you multiply 

3 + 5i
------
0 - 1i

by that:

3 + 5i   0 + 1i
------ · ------
0 - 1i   0 + 1i

Use FOIL in numerator and denominator

3·0 + 3·1i + 5i·0 + 5i2
-----------------------
0·0 + 0·1i - 0·1i - 1i2

 3i + 5i2
----------
  -1i2

Now we replace i2 by -1

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

 3i - 5
--------
    1

or just

 3i - 5

It is customary to write the "i" part last, so the answer is

-5 + 3i

Edwin
AnlytcPhil@aol.com