Show that is a rational number.
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Step-by-step explanation:
Hi,
Consider z = [( -1 + i√3)/2]³
= ( -1 /2+ i√3/2)³
Expanding using binomial expansion, we get
= ³C₀(-1/2)³ + ³C₁(-1/2)²(i√3/2) + ³C₂(-1/2)(i√3/2)² + ³C₃(i√3/2)³
On simplifying, we get
= -1/8 +3i√3/8 - 3*3i²/8 + (√3)³i³/8
But we know that i² = -1
i³ = i²*i = -1*i = -i
So , [( -1 + i√3)/2]³ = -1/8 + 3√3 i/8 -9(-1)/8 + 3√3*(-i)/8
= -1/8 + (3√3/8)i + 9 - (3√3/8)i
= 9/8 - 1/8
= 8/8
= 1 which is a rational number
Hence, [( -1 + i√3)/2]³ is a rational number.
Hope, it helps !
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