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How would you simplify 2/3-i?

I tried to multiply the 3-i on the top and bottom but then I get lost on what belongs on the bottom and on the top. 
 
I thought I would have something like 6-2i/9 but that doesn't seem right.

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Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...
4.8 4.8 (4 lesson ratings) (4)
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 This is division of 2 complex numbers.
 
 
      2__  =
     3 - i
   
 Multiply both numerator and denominator by conjugate number of the denominator.
 
    2     *   3+ i    =                   
 3 - i       3 +i                                                        
                                                                                 separate the rational and irrational numbers from
   each other                  
 
 
  6 + 2i   3  + 1  i            
    10           5     5
  
  This is also division of 2 complex numbers into each other.
Jason S. | My goal is the success of my students. Knowledge-Patience-HonestyMy goal is the success of my students. K...
4.9 4.9 (115 lesson ratings) (115)
0
 
Multiply top and bottom by the conjugate, 3+i:
 
2           3 + i
---   *    ------
3-i         3 + i
 
 
2(3+i)
--------
(3-i)(3+i)
 
6 + 2i
----------
9 + 3i - 3i - i^2 
 
6 + 2i
-------
9 - (-1)
 
6+2i
-------    or    (3/5) + (1/5) i
  10
 
(3/5) + (i/5) works as well.

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