Evaluate the quotient, and write the result in the form a + bi. (Simplify your answer completely.) 6 − 3i 1 − 6i

Answer 1

In order to solve a problem like this we have to multiply top and bottom by the conjugate of the denominator - or in simple terms, you multiply by the opposite of the bottom number.

So in your problem, we have

^6-3i/_1-6i

The conjugate of the denominator is 1+6i, so when we multiply top and bottom by 1+6i, the imaginary terms in the bottom will cancel out and no longer be an issue in the denominator

^6-3i/_1-6i * ¹⁺⁶ⁱ/_1+6i = ^(6-3i)(1+6i)/_(1-6i)(1+6i)

Now we can multiply by using FOIL (first, outer, inner, last)

^(6-3i)(1+6i)/_(1-6i)(1+6i) = ^{(6- 3i + 36i -18i^2)}/_{(1 - 6i + 6i -36i^2)} = ^{(6+33i - 18i^2)} / _{(1 - 36i^2)}

Since i = √-1, i² = -1. Now we can substitute -1 for i² in the above equation to get

^{(6+33i - 18i^2)} / _{(1 - 36i^2)} = ^{(6+33i - 18(-1))} / _{(1 - 36(-1))}

^{(24+ 33i)}/₃₇

It's customary to split up a number like this and write it as two fractions:

²⁴/₃₇ + ³³ⁱ/₃₇