Simplify the following:
abs((sqrt(3) i + 1)^4)
(sqrt(3) i + 1)^4 = ((sqrt(3) i + 1)^2)^2:
abs(((sqrt(3) i + 1)^2)^2)
(sqrt(3) i + 1)^2 = 1 + i sqrt(3) + i sqrt(3) - 3 = 2 i sqrt(3) - 2:
abs((2 i sqrt(3) - 2)^2)
Factor 2 out of 2 i sqrt(3) - 2 giving 2 (i sqrt(3) - 1):
abs((2 (i sqrt(3) - 1))^2)
(2 (i sqrt(3) - 1))^2 = 2^2 (i sqrt(3) - 1)^2:
abs(2^2 (i sqrt(3) - 1)^2)
abs(4 (i sqrt(3) - 1)^2)
(i sqrt(3) - 1)^2 = 1 - i sqrt(3) - i sqrt(3) - 3 = -2 i sqrt(3) - 2:
abs(4 -2 i sqrt(3) - 2)
Factor 2 out of -2 i sqrt(3) - 2 giving 2 (-i sqrt(3) - 1):
abs(4×2 (-(i sqrt(3)) - 1))
Factor -1 from -(i sqrt(3)) - 1:
abs(4×2×-(i sqrt(3) + 1))
abs(-8 (i sqrt(3) + 1))
abs(-8 (1 + i sqrt(3))) = abs(-8) abs(1 + i sqrt(3)):
abs(-8) abs(1 + i sqrt(3))
Since -8<=0, then abs(-8) = 8:
8 abs(1 + i sqrt(3))
abs(1 + i sqrt(3)) = 2:
= 16