#1**+1 **

See how the exponents are the same and so are both of the parts. Try to maybe simplify it. This is an answer you have to get using cheeky simplifying. See what you can cancel out, etc...

sarvajitjjjj Jun 26, 2020

#2**0 **

Notice that the complex numbers are \(\omega = \dfrac{-1 + i\sqrt 3}{2}\) and \(\omega^2 = \dfrac{-1 - i\sqrt 3}{2}\) respectively, where \(\omega\) is the cube root of unity.

The original expression equates to \(\omega^{2021} + \left(\omega^2\right)^{2021} = \omega^{2021} + \omega^{4042}\).

Noticing that \(\omega^3 = 1\), \(\omega^{2021} + \omega^{4042} = \omega^2 + \omega = \dfrac{-1-i\sqrt 3}2+ \dfrac{-1+i\sqrt 3}2 = \boxed{-1}\).

MaxWong Jun 26, 2020