Embibe Experts Solutions for Exercise 5: EXERCISE

$\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}=$

The roots of the equation ${\left(x-1\right)}^3+8=0$ are

If ${\left(\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)}^n$ is an integer, then $n$ is

${\left(1+2\omega +{\omega }^2\right)}^{3n}-{\left(1+\omega +2{\omega }^2\right)}^{3n}=$

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

If $x^2+x+1=0$, then find the value of $\sum _{r=1}^{27}{\left(x^r+\frac{1}{x^r}\right)}^2$.

Find the value of $4+5{\left(-\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)}^{334}+3{\left(-\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)}^{365}$.

If $\alpha =\cos \frac{2\pi }{5}+i\sin \frac{2\pi }{5}$, then find the value of $\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4$.