First slide

De-moivre's theorem

Question

If $1, ω, ω^{2}, \dots, ω^{n - 1}$ are n, nth roots of unity, the value of $(9 - ω) (9 - ω^{2}) \dots (9 - ω^{n - 1})$ will be

Moderate

A

n
B

0
C

$\frac{9^{n} - 1}{8}$
D

$\frac{9^{n} + 1}{8}$

Solution

$\begin{array}{l} let x = (1)^{1 / n} \\ x^{n} - 1 = 0 \\ or x^{n} - 1 = (x - 1) (x - ω) (x - ω^{2}) \dots (x - ω^{n - 1}) \\ \Rightarrow \frac{x^{n} - 1}{x - 1} = (x - ω) (x - ω^{2}) \dots (x - ω^{n - 1}) \end{array}$
Putting x = 9 in both sides, we have
$(9 - ω) (9 - ω^{2}) (9 - ω^{3}) \dots (9 - ω^{n - 1}) = \frac{9^{n} - 1}{8}$

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