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Find the smallest positive integer n such that ((sqrt(3) + i)/(1 +…
Question
Find the smallest positive integer n such that ((sqrt(3) + i)/(1 + i))^n is a real number. Report n.
✓ Verified answer: 12checked by our engine — not a guess
Step-by-step solution
Write each factor in polar form: sqrt(3) + i = 2*e^{i*pi/6} (modulus 2, argument 30 degrees) and 1 + i = sqrt(2)*e^{i*pi/4} (modulus sqrt2, argument 45 degrees).
Their quotient is (2/sqrt2)*e^{i(pi/6 - pi/4)} = sqrt(2)*e^{-i*pi/12}.
Raising to the n-th power gives modulus 2^{n/2} and argument -n*pi/12.
The result is real exactly when the argument is an integer multiple of pi: -n*pi/12 = m*pi, i.e.
n/12 is an integer.
The smallest positive such n is 12.
(At n = 12 the value is sqrt(2)^12 * e^{-i*pi} = 64*(-1) = -64, real.)
Final answer12
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