Basics of Complex Numbers

The value of the sum  $\sum _{n=1}^{13}\left(i^n\hspace{0.17em}+i^{n+1}\right),$ where  $i=\sqrt{-1}$ , equals

For positive integers  $n_1, n_2$  the value of expression ${\left(1+i\right)}^{n_1}+{\left(1+i^3\right)}^{n_1}+$${\left(1+i^5\right)}^{n_2}+{\left(1+i^7\right)}^{n_2}$ where  $i=\sqrt{-1}$ , is a real number if and only if

What is a complex number ?

Find the square root of $-196$ in imaginary unit $\text{'}i\text{'}$.

The square root of $-16$ is $4i$, where $i=\sqrt{-1}$.

Representation of the complex number $z=\left(-3-5i\right)$ in ordered pair form is $z:\left(-3, -5\right)$.

Representation of the complex number $z=\left(2-7i\right)$ in ordered pair form is $z:\left(2, 7\right)$.

Represent the complex number $z=\left(-1-i\right)$ in ordered pair form.

A complex number $z=2-4i$.

Representation of given complex number in ordered pair form is

A complex number is represented by an ordered pair $z:\left(2,5\right)$, which of the following is true for $z$?

If $n$ is an integer and $z=cis\theta$, then $\frac{z^{2n}-1}{z^{2n}+1}=$

If $x+iy=\frac{1}{1+\cos \theta +i\sin \theta }$, then $4x^2-1=$

If $1, \omega , {\omega }^2$ are three cube roots of unity, then ${\left(1-\omega +{\omega }^2\right)}^3=$

Find the value of ${\left[\frac{1+\sin {\displaystyle \frac{\pi }{8}}+i \cos {\displaystyle \frac{\pi }{8}}}{1+ \sin \frac{\pi }{8}-i \cos \frac{\pi }{8}}\right]}^{\frac{8}{3}}$

$(1-i{)}^8=$

If $x+iy=\frac{3}{2+\cos \left(\theta \right)+i\sin \left(\theta \right)},$ then $x^2+y^2=$________

A(z₁), B(z₂), C(z₃) are the vertices of a triangle ABC inscribed in the circle $\left|\text{z}\right|=2$. Internal angle bisector of the angle A meets the circum-circle again at D(z₄).

arg $\left[{\text{z}}_4/\left({\text{z}}_2-{\text{z}}_3\right)\right]$  is equal to

For two complex numbers $z_1=a-ib$ and $z_2=c-id$ then $z_1{z}_2$ is purely imaginary when 

If the complex number $\frac{2+3i}{4+5i}$ is equal to $\frac{a}{k}+i\frac{b}{k}$ then find the value of $a+b+k$.

If $(a+ib{)}^5=\alpha +i\beta ,$ then $(b+ia{)}^5=$