SOLUTION: perform the operation i/3-2i + 2i/3+8i and write the result in standard form

Question 222341: perform the operation i/3-2i + 2i/3+8i and write the result in standard form
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Since working with complex numbers uses many of the same techniques and facts as working with square roots. This includes the requirement for rational ("no square root") denominators. So when we see:

we should immediately recognize that those i's in the denominators are square roots and that, among other things, we will need to rationalize the denominators. In fact, since we cannot add the fractions (they have different denominators), then we'll start by rationalizing the denominators.

Rationalizing multiple term denominators like these is a little tricky. We have to use the pattern: . What is useful about this pattern is that it shows how to take a binomial (two term expression) like a+b or a-b and turn it into an expression with nothing but perfect square terms ().

Expressions like (a+b) and (a-b) are called "conjugates". So to rationalize a binomial denominator we will multiply the numerator and denominator by the conjugate of its denominator:

Continuing to simply:

Now we have rational denominators. Next we get common denominators so we can add:

To write this is standard a+bi form we just need to split this fraction:

(NOTE: These are pretty strange fractions. It doesn't mean they are wrong but they are suspicious since most problems in school are usually "nicer". I've checked for errors but you may want to do the same.)

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