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### Section 1.7 : Complex Numbers

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

$\frac{{6 + 7i}}{{8 - i}}$

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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 $$8 + i$$.

Show Step 2

Multiplying 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 3

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

$\frac{{6 + 7i}}{{8 - i}}\,\,\frac{{8 + i}}{{8 + i}} = \frac{{48 + 62i + 7{i^2}}}{{64 - {i^2}}} = \frac{{41 + 62i}}{{65}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{41}}{{65}} + \frac{{62}}{{65}}i}}$