<PRE><B>Question 24527</B>
&lt;pre&gt;Hi, can you show me how to solve a problem like this one?
  Find Multiplicative Inverse.

      -&lt;i&gt;i&lt;/i&gt;
     -----
      3+5&lt;i&gt;i&lt;/i&gt;


Is the answer just 3-5&lt;i&gt;i&lt;/i&gt;     ????
                  ------
                    &lt;i&gt;i&lt;/i&gt;         
&lt;b&gt;&lt;font size = 3&gt;
No.  You do invert it, but you don&#39;t change any signs yet.

It&#39;s simply the original problem inverted without changing any
signe

 
3 + 5&lt;i&gt;i&lt;/i&gt;
------
  -&lt;i&gt;i&lt;/i&gt; 

However that&#39;s not finished.

You write the denominator as 0 - 1&lt;i&gt;i&lt;/i&gt;

3 + 5&lt;i&gt;i&lt;/i&gt;
------
0 - 1&lt;i&gt;i&lt;/i&gt;

Then you form the conjugate of the denominator, 0 - 1&lt;i&gt;i&lt;/i&gt;.
(This is the only time where you change the sign of the second term.)
The conjugate of 0 - 1&lt;i&gt;i&lt;/i&gt; is 0 + 1&lt;i&gt;i&lt;/i&gt;.  So you form the fraction by putting
that over itself, so that it just equals 1

0 + 1&lt;i&gt;i&lt;/i&gt;
------
0 + 1&lt;i&gt;i&lt;/i&gt;

Then you multiply 

3 + 5&lt;i&gt;i&lt;/i&gt;
------
0 - 1&lt;i&gt;i&lt;/i&gt;

by that:

3 + 5&lt;i&gt;i&lt;/i&gt;   0 + 1&lt;i&gt;i&lt;/i&gt;
------ · ------
0 - 1&lt;i&gt;i&lt;/i&gt;   0 + 1&lt;i&gt;i&lt;/i&gt;

Use FOIL in numerator and denominator

3·0 + 3·1&lt;i&gt;i&lt;/i&gt; + 5&lt;i&gt;i&lt;/i&gt;·0 + 5&lt;i&gt;i&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt;
-----------------------
0·0 + 0·1&lt;i&gt;i&lt;/i&gt; - 0·1&lt;i&gt;i&lt;/i&gt; - 1&lt;i&gt;i&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt;

 3&lt;i&gt;i&lt;/i&gt; + 5&lt;i&gt;i&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt;
----------
  -1&lt;i&gt;i&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt;

Now we replace &lt;i&gt;i&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt; by -1

 3&lt;i&gt;i&lt;/i&gt; + 5(-1)
------------
   -1(-1)

 3&lt;i&gt;i&lt;/i&gt; - 5
--------
    1

or just

 3&lt;i&gt;i&lt;/i&gt; - 5

It is customary to write the &quot;&lt;i&gt;i&lt;/i&gt;&quot; part last, so the answer is

-5 + 3&lt;i&gt;i&lt;/i&gt;

Edwin
AnlytcPhil@aol.com&lt;/pre&gt;
</PRE>