Question

show that (root 3/2 + i/2)^3 is =i

guys plz help its urgent ...plz
and do remember it is whole cube not multiplied by 3....
help...​

Answer 1

$\underline{\textbf{To prove:}}$

$\mathsf{\left(\dfrac{\sqrt{3}}{2}+i\,\dfrac{1}{2}\right)^3=i}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Demovire's theorem:}}$

$\boxed{\mathsf{(cos\,\theta+i\,sin\theta)^n=cos\,n\theta+i\,sin\,n\theta}}$

$\mathsf{Consider,}$

$\mathsf{\left(\dfrac{\sqrt{3}}{2}+i\,\dfrac{1}{2}\right)^3}$

$\textsf{This can be written as,}$

$\mathsf{=\left(cos\dfrac{\pi}{6}+i\,sin\dfrac{\pi}{6}\right)^3}$

$\textsf{Using Demovire's theorem, we get}$

$\mathsf{=cos\dfrac{3\pi}{6}+i\,sin\dfrac{3\pi}{6}}$

$\mathsf{=cos\dfrac{\pi}{2}+i\,sin\dfrac{\pi}{2}}$

$\mathsf{=0+i\,1}$

$\mathsf{=i}$

$\implies\boxed{\mathsf{\left(\dfrac{\sqrt{3}}{2}+i\,\dfrac{1}{2}\right)^3=i}}$