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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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