Question

If $\text{ω}$ is imaginary cube root of unity then the value of $\frac{|3\omega -2{\omega }^2||1-\omega |}{|3{\omega }^2-2||{\omega }^2-1|}$ is

Answer 1

Answer: 1

We have,
$\frac{|3\omega -2{\omega }^2||1-\omega |}{|3{\omega }^2-2||{\omega }^2-1|}$
Multiply and divide by $\text{ω}$ , then
$=\frac{|3\omega -2{\omega }^2||\omega -{\omega }^2|}{|3{\omega }^3-2\omega ||{\omega }^2-1|}$
$=\frac{\omega |3-2\omega ||\omega -{\omega }^2|}{|3\cdot 1-2\omega ||{\omega }^2-1|}$
$=\frac{\omega |\omega -{\omega }^2|}{|{\omega }^2-1|}$
$=\frac{|{\omega }^2-{\omega }^3|}{|{\omega }^2-1|}$
$=\frac{|{\omega }^2-1|}{|{\omega }^2-1|}=1$