Question 421979
your expression is (4-2i) / (7 + 3i)


I initially thought this was a multiplication, but I now see it's a division.


you would want to remove the i terms from the denominator.


if you multiply both numerator and denominator by (7-3i), that should do it.


your expression becomes:


(4-2i) * (7-3i) divided by (7+3i) * (7-3i)


(7+3i) * (7-3i) would be equal to 49 - 21i + 21i - 9i^2


the middle terms cancel out and you are left with 49 -9i^2


the i's are treated like any other variable until the end, at which time they are finally processed based on the rules for i processing to be described after all is said and done.


anyway, your denominator is 49 - 9i^2


you would multiply your numerator out to get:


(4-2i) * (7-3i) = 28 - 12i - 14i + 6i^2


combine like terms to get your numerator equal to 28 - 26i + 6i^2


your expression now has become:


28 - 26i + 6i^2 divided by 49 - 9i^2


now you process the i's based on the rules.


i = i
i^2 = -1
i^3 = -i
i^4 = 1


this is a cyclical pattern that repeats every 4 exponents.


i^5 = i
i^6 = -1
i^7 = -i
i^8 = 1


i^9 = i
i^10 = -1
i^11 = -i
i^12 = 1


etc.


your expression is, once again:


28 - 26i + 6i^2 divided by 49 - 9i^2


since i^2 is equal to -1, you change your expression to become:


28 - 26i + 6*(-1) divided by 49 - 9*(-1)


this simplifies to:


28 - 26i - 6 divided by 49 + 9


combine like terms to get:


22 - 26i divided by 58


divide both numerator and denominator by 2 to get:


11 - 13i divided by 29


that should be your answer if i did it right.


here's a decent reference that pretty much tells you the same thing.


<a href = "http://www.sparknotes.com/math/algebra2/complexnumbers/section3.rhtml" target = "_blank">http://www.sparknotes.com/math/algebra2/complexnumbers/section3.rhtml</a>


you remove the complex number from the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.


if the denominator is a + bi, then the conjugate is a - bi


if the denominator is a - bi, then the conjugate is a + bi


both the numerator and the denominator have to be multiplied by the same factor in order to preserve the integrity of the expression.


2/4 * 2/2 = 4/8


the integrity of the expression is maintained because 4/8 is equivalent to 2/4.


between what i told you and the reference, you should get the idea.