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Evaluate the complex expression ((1)/(1-2i)+(3)/(1+i)) ((3+4i)/(2-4i)):

1. (1)/(2)+(9)/(2)i
2. (1)/(2)-(9)/(2)i
3. (1)/(4)-(9)/(4)i
4. (1)/(4)+(9)/(4)i

Correct answer: (1)/(4)+(9)/(4)i

Solution

1/(1-2i) = (1+2i)/5 and 3/(1+i) = 3(1-i)/2; their sum is (2/10+4i/10)+(15/10-15i/10) = (17 -11i)/10. Multiplying by (3+4i)/(2-4i) = (3+4i)(2+4i)/20 = (-10+20i)/20 = (-1+2i)/2 gives (17-11i)(-1+2i)/20 = (5+45i)/20 = 1/4 + (9/4)i.

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