Question

Let $\alpha ,\beta$ be the imaginary cube roots of unity. Then the value of $\left({\alpha }^4{\beta }^4\right)-\frac{1}{\alpha \beta }$.

• A. $1$
• B. $2$
• C. $0$
• D. none of these

Answer 1

The correct option is C $0$

We have the imaginary cube roots of unity are $\omega ,{\omega }^2$ and ${\omega }^3=1$.

Now let $\alpha =\omega$ or ${\omega }^2$ then $\beta ={\omega }^2$ or $\omega$.

For both the cases $\alpha .\beta ={\omega }^3=1$......(1).

Now,

$\left({\alpha }^4{\beta }^4\right)-\frac{1}{\alpha \beta }$

$=(\alpha \beta {)}^4-\frac{1}{\alpha \beta }$

$=1-1$ [ Using (1)]

$=0$.