Question 164498
What does the quotient equal?
{{{(3+i)/(2-3i)}}}
<pre><font size = 4 color = "indigo"><b>
Form the conjugate of the denominator:

The denominator is {{{2-3i}}} so to form
the conjugate we use the first term but
we change the sign of the term in {{{i}}},
so the conjugate of {{{2-3i}}} is {{{2+3i}}}
Put that over itself. That is, we form the 
fraction {{{(2+3i)/(2+3i)}}} which is just {{{1}}} 
in value.

Then we multiply the original expression by
this:

{{{matrix(1,3, (3+i)/(2-3i), "×" , (2+3i)/(2+3i) )}}}

Put parentheses around everything:

{{{matrix(1,3, ((3+i))/((2-3i)), "×" , ((2+3i))/((2+3i)) )}}}

Indicate the multiplication of numerators and denominators:

{{{((3+i)(2+3i))/((2-3i)(2+3i))}}}

Use FOIL on the top and bottom:

{{{(6+9i+2i+3i^2)/(4+6i-6i-9i^2)}}}

{{{(6+11i+3i^2)/(4+cross(6i)-cross(6i)-9i^2)}}}

{{{(6+11i+3i^2)/(4-9i^2)}}}

Now replace the {{{i^2}}}'s by {{{-1}}}

{{{(6+11i+3(-1))/(4-9(-1))}}}

{{{(6+11i-3)/(4+9)}}}

{{{(3+11i)/13}}}

Make two fractions:

{{{3/13 + 11i/13}}}

To write it in the form {{{A+Bi}}},

{{{3/13 + 11/13}}}{{{i}}}

Edwin</pre>