The number of real values of (-1-sqrt{3} i)^{ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let $z = -1 - \sqrt{3}i$. First,express $z$ in polar form: $|z| = \sqrt{(-1)^2 + (-\sqrt{3})^2} = \sqrt{1+3} = 2$. $	heta = 	ext{arg}(z) = \pi +

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(C) Let $z = -1 - \sqrt{3}i$. First,express $z$ in polar form: $|z| = \sqrt{(-1)^2 + (-\sqrt{3})^2} = \sqrt{1+3} = 2$. $	heta = 	ext{arg}(z) = \pi + 	an^{-1}(\frac{\sqrt{3}}{1}) = \pi + \frac{\pi}{3} = \frac{4\pi}{3}$. So,$z = 2e^{i(4\pi/3 + 2k\pi)}$ for $k = 0, 1, 2, 3$. Then $z^{3/4} = (2e^{i(4\pi/3 + 2k\pi)})^{3/4} = 2^{3/4} e^{i(3/4)(4\pi/3 + 2k\pi)} = 2^{3/4} e^{i(\pi + 3k\pi/2)}$. For the value to be real,the imaginary part must be zero,i.e.,$\sin(\pi + \frac{3k\pi}{2}) = 0$. This implies $\pi + \frac{3k\pi}{2} = n\pi$ for some integer $n$. Dividing by $\pi$,we get $1 + \frac{3k}{2} = n$,or $2 + 3k = 2n$. Since $2n$ is even and $2$ is even,$3k$ must be even,which means $k$ must be even. For $k \in \{0, 1, 2, 3\}$,the even values are $k = 0$ and $k = 2$. For $k=0$,$	ext{arg} = \pi$,$e^{i\pi} = -1$ (real). For $k=2$,$	ext{arg} = \pi + 3\pi = 4\pi$,$e^{i4\pi} = 1$ (real). Thus,there are $2$ real values.

The number of real values of $(-1-\sqrt{3} i)^{3/4}$ is

Similar Questions

If $a = \sqrt{2i}$,then which of the following is correct?

If $\theta = \frac{\pi}{6}$,then the $10^{th}$ term of the series $1 + (\cos \theta + i \sin \theta) + (\cos \theta + i \sin \theta)^2 + (\cos \theta + i \sin \theta)^3 + \ldots$ is equal to:

If $\alpha$ is a non-real root of the equation $x^6-1=0$,then $\frac{\alpha^2+\alpha^3+\alpha^4+\alpha^5}{\alpha+1} = $

If $\omega$ is a complex cube root of unity,then the value of $\omega^{99} + \omega^{100} + \omega^{101}$ is

The product $(1 - \omega + {\omega ^2})(1 - {\omega ^2} + {\omega ^4})(1 - {\omega ^4} + {\omega ^8}) \dots$ up to $2n$ factors is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module