SOLUTION: Please help me simplify this equation: (4-2i)/(7+3i)

Question 421979: Please help me simplify this equation: (4-2i)/(7+3i)
Found 2 solutions by shree840, Theo:
Answer by shree840(260) (Show Source): You can put this solution on YOUR website!
(4-2i)(7+3i)
remember i^2=-1
use either foil method or break it up
f first 4*7=28
o outer 4*3i=12i
i innner -2i*7=-14i
l last -2i*3i=-6i^2 =-6*-2=12
28+12i-14i+12
-2i+40
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
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.

http://www.sparknotes.com/math/algebra2/complexnumbers/section3.rhtml

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.

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