SOLUTION: assignment is to simplify this expression: (3-i)^-2 i get as far as 1/(9-6i+i^2) answers sheet says correct answer is (4+3i)/50 can you help me figure out how to get that

Algebra ->  Exponents-negative-and-fractional -> SOLUTION: assignment is to simplify this expression: (3-i)^-2 i get as far as 1/(9-6i+i^2) answers sheet says correct answer is (4+3i)/50 can you help me figure out how to get that      Log On


   

Question 78176: assignment is to simplify this expression: (3-i)^-2
i get as far as 1/(9-6i+i^2)
answers sheet says correct answer is (4+3i)/50
can you help me figure out how to get that answer?

Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%283-i%29%5E-2 Start with the given expression
1%2F%283-i%29%5E2 Invert
1%2F%289-6i%2Bi%5E2%29 Foil
1%2F%289-6i%2B-1%29 replace i%5E2 with (-1) Since i=sqrt%28-1%29 i%5E2=-1
1%2F%288-6i%29 Combine like terms
%281%2F%288-6i%29%29%28%288%2B6i%29%2F%288%2B6i%29%29 Multiply the numerator and denominator by the complex conjugate 8%2B6i this will rationalize the denominator (ie it will get rid of the i term in the denominator)
%288%2B6i%29%2F%28%288-6i%29%288%2B6i%29%29 Multiply
%288%2B6i%29%2F%2864%2Bcross%2848i-48i%29-36i%5E2%29 Foil
%288%2B6i%29%2F%2864-36i%5E2%29 Combine like terms
%288%2B6i%29%2F%2864-36%28-1%29%29 Replace i%5E2 with (-1)
%288%2B6i%29%2F%2864%2B36%29 Multiply -36 by -1
%288%2B6i%29%2F100Combine like terms
2%284%2B3i%29%2F100 Factor out a 2
%284%2B3i%29%2F50 Divide

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
simplify this expression: (3-i)^-2 = 1/(3-i)^2
Multiply numerator and denominator by (3+i)^2 to get:
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(3+i)^2 / 10^2
= (9+6i-1)/100
= (8+6i)/100
= (4+3i)50
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