Question 345278
<pre><font size = 3 color = "indigo"><b>
{{{(5+i)/(2+9i)}}}

The denominator is {{{2+9i}}}.  Form the conjugate by
changing the sign only the term that contains the letter
i.

So the conjugate of {{{red(2+9i)}}} is {{{red(2-9i)}}}

Now put that conjugate over itself: {{{red((2-9i)/(2-9i))}}} 

That just equals 1 because any number (other than 0) divided by
itself is 1.  So we multiply the original problem by that:

{{{expr((5+i)/(2+9i))*red(expr((2-9i)/(2-9i)))}}}

{{{((5+i)red((2-9i)))/((2+9i)red((2-9i)))}}}

Use FOIL on the top and the bottom:

{{{(10-45i+2i-9i^2)/(4-18i+18i-81i^2)}}}

Simplify:

{{{(10-43i-9i^2)/(4-81i^2)}}}

Now use the fact that since {{{i=sqrt(-1)}}}, the {{{i^2=-1}}}
So we replace {{{i^2}}} by {{{-1}}}

{{{(10-43i-9(-1))/(4-81(-1))}}}

{{{(10-43i+9)/(4+81)}}}

{{{(19-43i)/85}}}

Make two fractions

{{{19/85-expr(43/85)i}}}

Edwin</pre>