InterviewSolution
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InterviewSolution
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Write the comple number in a + ib form unsing cube roots of unity: (a) `(-(1)/(2) + sqrt(3)/(2)i)^(1000)`(b)If `z =(sqrt(3) + i)^(17)/((1-i)^(50))` (c) `(i + sqrt(3))^(100) + (i+ sqrt(3))^(100) + 2^(100)` |
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Answer» Correct Answer - `omega = - (1)/(2) + sqrt(3)/(2)i`" " (b) `(1)/(2^(8))((-1-isqrt(3))/(2))` " "(c) 0 (a) Here, `-1//2 + (1//2) isqrt(3)` is one of the tow imaginary cube roots of unity. If the we denote it b `omega`, then `omega^(100) = omega^(999) omega=(omega^(3))^(333) omega = omega = -(1)/(2) + (sqrt(3))/(2) i` `z =(sqrt(3) + i)^(17)/((1-i)^(50))` `= (1)/(i^(17))((isqrt(3) +i^(2))^(17))/([(1-i)^(2) ]^(25))` `= (2^(17))/(i) (((isqrt(3)-1)/(2))^(7))/(-2i)^(25)` `=(1)/(2^(8)) (omega)^(17)` `= (1)/(2^(8)) (omega)^(2)` `= (1)/(2^(8)) ((-1-isqrt(3))/(2))` (c) `(i+sqrt(3))^(100) + (i-sqrt(3))^(100) + 2^(100)` `=((i^(2) + isqrt(3))/(i))^(100) + ((i^(2)-isqrt(3))/(i))^(100) + 2^(100)` `= (2^(100))/(i^(100)) ((-1+isqrt(3))/(2))^(100) + (2^(100))/(i^(100)) ((-1-isqrt(3))/(2)) +2^(100)` `= 2^(100) (omega) ^(100) + 2^(100) (omega^(2))^(100) + 2^(100)` `=2^(100) (omega^(100) + omega^(200)+1)` ` = 2^(100) (omega + omega^(2) + 1) =0` |
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