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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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