Video Transcript
Evaluate nine minus 𝜔 squared plus
nine 𝜔 to the fourth power all squared plus six plus six 𝜔 squared plus six 𝜔 to
the fourth power all squared, where 𝜔 is a nontrivial cubic root of unity.
In this question, we are asked to
evaluate an expression involving 𝜔, which we are told is a nontrivial cubic root of
unity.
To answer this question, we can
start by noting that since 𝜔 is a cube root of unity, the cube of 𝜔 must be equal
to one. We can multiply this equation
through by 𝜔 to see that 𝜔 to the fourth power is equal to 𝜔. We can use this to rewrite the
given expression. We want to replace every instance
of 𝜔 to the fourth power with 𝜔. Doing this gives us the following
expression. We can simplify the expression
further by recalling that the sum of the powers of a nontrivial root of unity is
zero. So one plus 𝜔 plus 𝜔 squared
equals zero.
We can take out the shared factor
of six in the second term to obtain the following expression. We can then note that the factor of
one plus 𝜔 plus 𝜔 squared is equal to zero. Therefore, this term is equal to
zero, so we are left with only the first term. We can simplify even further by
noting that there are two terms which share a factor of nine. We can then rewrite this factor in
terms of 𝜔 squared by noting that one plus 𝜔 must be equal to negative 𝜔
squared. We can substitute this into the
expression to get nine times negative 𝜔 squared minus 𝜔 squared all squared. We can then simplify the expression
inside the parentheses to obtain negative 10𝜔 squared all squared.
Now, we apply the laws of exponents
to take the power of each factor separately. We have negative 10 squared times
𝜔 squared squared, which is equal to 𝜔 to the fourth power. Finally, we can use the fact that
𝜔 to the fourth power is equal to 𝜔 to simplify the expression to obtain a final
answer of 100𝜔.
