Show that `(-1+sqrt(-3i))^3` is a real number

Show that 2-sqrt(3) is an irrational number. Given sqrt(3) is irrational.

Show that (2 + sqrt(3)) is an irrational number.

Show that 5-2sqrt(3) is an irrational number.

Show that (sqrt3/2 + i/2)^3 = i

Show that 3+sqrt(2) is an irrational number.

Show that 1+i^(10)+i^(20)+i^(30) is a real number.

Prove that (3+i)/(1+2i)+(3-i)/(1-2i) is a real number.

Show that the square to (sqrt(26-15sqrt(3)))//(5sqrt(2)-sqrt(38+5sqrt(3))) is a rational number.

Let (-2 -1/3 i)^3=(x+iy)/27 (i= sqrt(-1)) where x and y are real numbers then y-x equals

If (-2 + sqrt-3) (-3 + 2 sqrt-3)=a +bi , find the real numbers a and b with values of a and b, also find the modulus of a+bi