Section 1.7 : Complex Numbers
9. Perform the indicated operation and write your answer in standard form.
\[\frac{{6 + 7i}}{{8 - i}}\]Show All Steps Hide All Steps
Hint : Recall that standard form does not allow any \(i\)'s in the denominator.
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 \(8 + i\).
Show Step 2Multiplying by the conjugate gives,
\[\frac{{6 + 7i}}{{8 - i}}\,\,\frac{{8 + i}}{{8 + i}} = \frac{{\left( {6 + 7i} \right)\left( {8 + i} \right)}}{{\left( {8 - i} \right)\left( {8 + i} \right)}}\] Show Step 3Now all we need to do is do the multiplication in the numerator and denominator and put the result in standard form.
\[\frac{{6 + 7i}}{{8 - i}}\,\,\frac{{8 + i}}{{8 + i}} = \frac{{48 + 62i + 7{i^2}}}{{64 - {i^2}}} = \frac{{41 + 62i}}{{65}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{41}}{{65}} + \frac{{62}}{{65}}i}}\]