Question

If \[1,\omega ,{{\omega }^{2}}\] are the cube roots unity then, \[(1+\omega )(1+{{\omega }^{2}})(1+{{\omega }^{4}})(1+{{\omega }^{8}})\] is equal to
1. \[1\]
2. \[0\]
3. \[\omega \]
4. \[{{\omega }^{2}}\]

Answer 1

Hint: To solve this question you need to know what are cube roots of unity and how to solve them therefore need to know important characteristics of them like these formula \[1+\omega +{{\omega }^{2}}=0\] and \[{{\omega }^{3}}=1\]. We must also know the basic rules of maths and the distribution formulas to be able to further simplify this question and solve it. In the end we will apply the necessary calculations needed to finish solving this and get the answer needed.

Complete step-by-step answer:
Now in the question it is said to us how \[1,\omega ,{{\omega }^{2}}\] are the cube roots of unity so we need to use that to solve this question that’s why we first need to know that if \[1,\omega ,{{\omega }^{2}}\] are cube roots of unity then
\[1+\omega +{{\omega }^{2}}=0\]
Therefore this means that
\[1+{{\omega }^{2}}=-\omega \]
Also
\[1+\omega =-{{\omega }^{2}}\]
Now that we know this we can substitute the values of these equations into our expression to then be able to further simplify this.
\[(1+\omega )(-\omega )(1+\omega )(1+{{\omega }^{2}})\]
Now we can write this as
\[(1+\omega )(1+\omega )(-\omega -{{\omega }^{3}})\]
Further simplifying using cube roots of unity’s formulas we get that
\[(1+\omega )(1+\omega )(-\omega -1)\]
We can write \[1+\omega \] in the form of the formula we derived which is this \[1+\omega =-{{\omega }^{2}}\] further simplifying the expression
\[{{(-{{\omega }^{2}})}^{2}}(-\omega -1)\]
Likewise we can write the second bracket in different form too, ti further simplify
\[{{(-{{\omega }^{2}})}^{2}}({{\omega }^{2}})\]
Opening the brackets squaring and multiplying we get
\[{{\omega }^{6}}\] which is equal to \[1\]. Therefore the answer for this expression we need is \[1\] that is the option \[1\] in this question.
So, the correct answer is “Option 1”.

Note: Cube roots of unity means the cube roots of 1 that is these are the three roots of the equation \[{{x}^{3}}-1=0\]. These roots are very useful for simplifying the solutions when we use huge complex numbers because these roots simplify how we will depict the complex numbers.