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Simplify 

 Jun 26, 2020
 #1
avatar+6 
+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...

 Jun 26, 2020
 #2
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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 \(\omega^3 = 1\)\(\omega^{2021} + \omega^{4042} = \omega^2 + \omega = \dfrac{-1-i\sqrt 3}2+ \dfrac{-1+i\sqrt 3}2 = \boxed{-1}\).

 Jun 26, 2020
 #3
avatar+8340 
0

The details are left to you.

MaxWong  Jun 26, 2020

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