Question

Which is the following is the square root of the following

Answer 1

`\displaystyle √(3)\biggr[\cos\biggr((11\pi)/(9)\biggr)+i\sin\biggr((11\pi)/(9)\biggr)\biggr]`

Explanation:

Use the nth-root formula for complex numbers in polar form

`\displaystyle \sqrt[n]{r}\biggr[\text{cis}\biggr((\theta+2\pi k)/(n)\biggr)\biggr]\\ \\\\\sqrt[2]{3}\biggr[\text{cis}\biggr(((4\pi)/(9)+2\pi(1))/(2)\biggr)\biggr]\\ \\\\√(3)\biggr[\text{cis}\biggr((2\pi)/(9)+\pi\biggr)\biggr]\\\\\\√(3)\biggr[\text{cis}\biggr((11\pi)/(9)\biggr)\biggr]`

`\displaystyle √(3)\biggr[\cos\biggr((11\pi)/(9)\biggr)+i\sin\biggr((11\pi)/(9)\biggr)\biggr]`

Note that this is one of the square roots of the complex number with k = n-1 = 2-1 = 1, but is the only applicable answer to this problem.