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### Section 1-7 : Complex Numbers

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

$\frac{{1 + 5i}}{{ - 3i}}$

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Hint : Recall that standard form does not allow any $$i$$'s in the denominator.
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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 $$3i$$.

Show Step 2

Multiplying by the conjugate gives,

$\frac{{1 + 5i}}{{ - 3i}}\,\,\frac{{3i}}{{3i}} = \frac{{\left( {1 + 5i} \right)\left( {3i} \right)}}{{\left( { - 3i} \right)\left( {3i} \right)}}$ 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.

$\frac{{1 + 5i}}{{ - 3i}} = \frac{{3i + 15{i^2}}}{{ - 9{i^2}}} = \frac{{ - 15 + 3i}}{9} = - \frac{{15}}{9} + \frac{3}{9}i = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{5}{3} + \frac{1}{3}i}}$