Complex Numbers

Find the value of $\frac{i^{9096}+i^{4397}+i^{2941}+i^{7986}}{i^{7976}+i^{4387}+i^{2931}+i^{5086}}$

If $z=2+3i, \text{then} z^5+{\left(\overline{z}\right)}^5$ is equal to:

If ${\left(1+x+x^2\right)}^{6n}=a_0+a_{1}x+a_2{x}^2+......,\text{then} a_0+a_3+a_6+.....=$

$\mathrm{If} \mathrm{a}=\cos \mathrm{\alpha }+\mathrm{isin}\mathrm{\alpha }, \mathrm{b}={\mathrm{e}}^{\mathrm{i\beta }}, \mathrm{c}=\mathrm{cis}\mathrm{\theta } \mathrm{and} \frac{\mathrm{ab}}{\mathrm{c}}=1, \mathrm{then} \mathrm{find} \mathrm{the} \mathrm{value}of (\mathrm{\alpha }+\mathrm{\beta }-\mathrm{\theta })$

If $\omega$ is an imaginary cube root of unity, then find the value of $\left(1+\omega \right)\left(1+{\omega }^2\right)\left(1+{\omega }^3\right)\left(1+{\omega }^4\right)\left(1+{\omega }^5\right).........\left(1+{\omega }^{3n}\right).$

Find the value of $\frac{1+i}{1-i}$

If $\left|z\right|=1$ and $\mathrm{z}\text{'}=\frac{1+{\mathrm{z}}^2}{\mathrm{z}}$ then,

Which of the following is correct ?

Express $i^{-35}$ in the form of $a+bi$.

Can we compare $2i$ and $i$? $i=\sqrt{-1}$

$\text{If }{\mathrm{z}}_{\mathrm{k}}=\cos \left(\frac{\mathrm{k}\pi }{10}\right)+\mathrm{isin}\left(\frac{\mathrm{k}\pi }{10}\right)\text{, then }{\mathrm{z}}_1{\mathrm{z}}_2{\mathrm{z}}_3{\mathrm{z}}_4=$

If $g\left(x\right)=x^4-x^3+x^2+3x-5$, then find $g\left(2+3i\right)$

The conjugate of $\frac{7+i}{10}$ in the form of $a+ib$ is

Find all possible solutions of $z^2=\overline{z},$ where $z\in C$.

Find the value of $i^{37}+\frac{1}{i^{67}}$

If $1,\omega ,{\omega }^2,.......,{\omega }^{n-1}$ are $n, n^{th}$ roots of unity, the value of $\left(7-\omega \right)\left(7-{\omega }^2\right).....\left(7-{\omega }^{n-1}\right)$ equals 

Express the following in the form of $a+bi$:

$(-5i)\left(\frac{1}{8}i\right)$

If $x=\frac{-5+i\sqrt{3}}{2},$ the prove that: 

${\left(x^2+5x\right)}^2+x\left(x+5\right)=42.$

Express the following in $a+ib$ form: $-5\left(\frac{1}{8}i\right)$

Evaluate $i^9+i^{19}$