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Section 1-7 : Complex Numbers

7. Perform the indicated operation and write your answer in standard form.

\[\frac{{7 - i}}{{2 + 10i}}\]

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Hint : Recall that standard form does not allow any \(i\)'s in the denominator.
Start Solution

Because standard form does not allow for \(i\)’s to be in the denominator we’ll need to multiply the numerator and denominator by the conjugate of the denominator, which is \(2 - 10i\).

Show Step 2

Multiplying by the conjugate gives,

\[\frac{{7 - i}}{{2 + 10i}}\,\,\frac{{2 - 10i}}{{2 - 10i}} = \frac{{\left( {7 - i} \right)\left( {2 - 10i} \right)}}{{\left( {2 + 10i} \right)\left( {2 - 10i} \right)}}\] Show Step 3

Now all we need to do is do the multiplication in the numerator and denominator and put the result in standard form.

\[\frac{{7 - i}}{{2 + 10i}} = \frac{{14 - 72i + 10{i^2}}}{{4 - 100{i^2}}} = \frac{{4 - 72i}}{{104}} = \frac{4}{{104}} - \frac{{72}}{{104}}i = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{{26}} - \frac{9}{{13}}i}}\]