find the value of #10

The two expressions within the brackets are the two complex cube roots of unity, $\omega, \omega^2$ with $1+\omega+\omega^2=0,\omega^3=1$

The given expression equals $\omega^2+\omega=\boxed {-1}$ .

Let z1 and z2 be the numbers in the first and second brackets respectively. Using polar form, z1 = exp(2 Pi (i)/3) and z2 = exp(4 Pi (i)/3)). Raising each to the 2021st power we find (z1)^2021 = exp(4042 Pi (i)/3) = exp(4 Pi (i)/3) = -1/2 - i sqrt(3) / 2 (the penultimate equality follows from the periodicity of the sine and cosine). Similarly, (z2)^2021 = -1/2 + i sqrt(3)/2, so that the sum of the two numbers is -1/2 - 1/2 = -1.