Algebra of Complex Numbers

Let $S_k=\left\{z\in C:z^2+\left|z\right|=k\right\}$ for each $k\in Z$. Then

If $z_1=-2-\sqrt{3}i$ and $z_2=-5-2\sqrt{3}i$  then the difference of two complex numbers in euler form is_________.

If $z_1=2+3i$ and $z_2=5+6i$ then the difference of two complex numbers in euler form is_________.

Let $z_1=a+ib$ and $z_2=c+id$ be complex numbers such that $\left|z_1\right|=\left|z_2\right|=1$ and $Re\left(z_1\overline{z_2}\right)=0$. Then the complex numbers $w_1=a+ic$ and $w_2=b+id$ always satisfy

Calculate $\log z$ for $z=-1-\sqrt{3}i$..

If $z_1$ and $z_z$ are any two complex numbers, then $\mathrm{Re}\left(z_1{z}_2\right)$ is

The equation $z\overline{z}+2\left(z+\overline{z}\right)-1=0$ represents

$\cos \left(i\log \left(\frac{a-ib}{a+ib}\right)\right)$ is equal to

The value of $\left|\sqrt{2i}-\sqrt{-2i}\right|$ is

If $x^2+y^2=1$, then the value of $\frac{1+x+iy}{1+x-iy}$ is

Subtract the complex number $2+4i$ from the complex number $5+i$ in the vector form.

Subtract the complex number $2+4i$ from the complex number $5+2i$ in the vector form.

Subtract the complex number $2+3i$ from the complex number $5+2i$ in the vector form.

Subtract the complex number $1+3i$ from the complex number $5+2i$ in the vector form.

Subtract the complex number $1+3i$ from the complex number $4+2i$ in the vector form.

Find the sum of the given complex numbers in the vector form.

$Z_1=2+3i$, and $Z_2=3-i$

Find the sum of the given complex numbers in the vector form.

$z_1=3+2i$, and $z_2=3-i$

Find the sum of the given complex numbers in the vector form.

$Z_1=3+3i$, and $Z_2=3-i$

Find the sum of the given complex numbers in the vector form.

$Z_1=3+4i$, and $Z_2=4-i$

Find the sum of the given complex numbers in the vector form.

$Z_1=-1+3i$, and $Z_2=4-i$