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(1+w+w^2)=

(complex number)

Asked by Ishit | 10 May, 2020, 10:48: AM
Expert Answer
To find: (1+w-w2)7
For a complex number w(omega),
w3 = 1
w3 - 1 = 0
Using the cubic identity, we have
open parentheses omega minus 1 close parentheses open parentheses 1 plus omega plus omega squared close parentheses equals 0
S i n c e comma space omega not equal to 1
therefore space 1 plus omega plus omega squared equals 0 space... space left parenthesis i right parenthesis
N o w comma space c o n s i d e r
open parentheses 1 plus omega minus omega squared close parentheses to the power of 7
equals open parentheses negative omega squared minus omega squared close parentheses to the power of 7 space... space U sin g space left parenthesis i right parenthesis
equals open parentheses negative 2 omega squared close parentheses to the power of 7
equals negative 2 to the power of 7 omega to the power of 14
equals negative 128 omega to the power of 12 cross times omega squared
equals negative 128 omega squared space space... space open parentheses S i n c e space omega cubed equals 1 space rightwards double arrow space omega to the power of 12 equals open parentheses omega cubed close parentheses to the power of 4 equals 1 close parentheses
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