Question

if (1+√3i)÷(1-√3i)^n is an integer then n is​

Answer 1

Answer:

n is a multiple of 3

Step-by-step explanation:

Presuming the question actually means

[ ( 1 + √3i ) / ( 1 - √3i ) ]ⁿ

i.e. the whole thing raised to the power of n, not just the denominator.

Let w = ( 1 + √3i ) / ( 1 - √3i )

Multiplying numerator and denominator by ( 1 + √3i ) gives

w = ( 1 + √3i )² / [ ( 1 - √3i ) ( 1 + √3i ) ]

  = ( 1 - 3 + 2√3i ) / ( 1 + 3 )

  = ( -2 + 2√3i ) / 4

  = ( -1 + √3i) / 2.

So w is a primitive cubed root of unity.     (*) ... see note below

So wⁿ is an integer

<=> wⁿ = 1

<=> n is a multiple of 3.

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(*)  If you don't recognize this, then take the conjugate w' = ( -1 - √3i) / 2 and

notice that

• w + w' = -1/2 + -1/2 = -1
• ww' = ( 1 + 3 ) / 2² = 1

=> w and w' are roots of x²+x+1, and therefore also of (x-1)(x²+x+1) = x³-1.