Question 315734
Write the following expressions as a complex number in standard form.
1).  {{{(4+5i)/(6+i)}}}
<pre><b>
Form the conjugate of {{{6+i}}} by changing the sign of the term
containing {{{i}}} and leaving the sign of the first term.

Thus the conjugate of {{{6+i}}} is {{{6-i}}}.

Place that conjugate over itself, like this {{{((6-i)/(6-i))}}},
which just equals 1, and so we can now multiply the original
expression by that without changing its value:

{{{((4+5i)/(6+i))}}}{{{((6-i)/(6-i))}}}

{{{( (4+5i)(6-i) ) / ( (6+i)(6-i) )}}}

Using FOIL on top and bottom:

{{{ (24-4i+30i-5i^2)/(36-6i+6i-i^2)}}}

Combining like terms (the middle terms cancel in the bottom)

{{{ (24+26i-5i^2)/(36-i^2)}}}

Replace {{{i^2}}} by {{{-1}}}

{{{ (24+26i-5(-1))/(36-(-1))}}}

Simplify:

{{{ (24+26i+5)/(36+1)}}}

{{{ (29+26i)/37}}}

Make two fractions, and write the {{{i}}} as multiplied on the right of
the second fraction, so it will be in the standard form {{{A+Bi}}}:

{{{29/37}}}{{{"+"}}}{{{26/37}}}{{{i}}}

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2). {{{14/i}}}

You only need to multiply that by {{{i/i}}}

{{{(14/i)}}}{{{(i/i)}}}

{{{(14i)/i^2}}}

Replace the {{{i^2}}} by {{{-1}}}

{{{(14i)/(-1)}}}

{{{-14i}}}

Edwin</pre>