Question

Show that $z=\Big(\frac{-1+\sqrt{-3}}{2}\Big)^{3}$ is a rational number.

Answer 1

Answer:

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 !