Logo
Logout

Roots of Complex Numbers - Questions

Question 1
21641

Find the four roots of \(\sqrt 3 + i\)

\({z_1} = {2^{\dfrac{1}{4}}}cis\dfrac{\pi }{{24}}\)

\({z_2} = {2^{\dfrac{1}{4}}}cis\dfrac{{13\pi }}{{24}}\)

\({z_3} = {2^{\dfrac{1}{4}}}cis\left( {\dfrac{{ - 11\pi }}{{24}}} \right)\)

\({z_4} = {2^{\dfrac{1}{4}}}cis\left( {\dfrac{{ - 23\pi }}{{24}}} \right)\)

Question 2
17255

The cube roots of \(i\) are ?

\({z_1} = cis\left( { \dfrac{\pi }{6}} \right),{z_2} = cis\left( { \dfrac{{5\pi }}{6}} \right),{z_3} = cis\left( {-\dfrac{\pi }{2}} \right)\)

Question 3
12821

The cube roots of 1 are?

\(1,\dfrac{{ - 1 \pm i\sqrt 3 }}{2}\)

Question 4
12824

The cube roots of \({\rm{2}} + 2i\) in mod - arg form are?

\({z_1} = \sqrt 2 {\rm{cis}}\left( {\dfrac{\pi }{{12}}} \right),{z_2} = \sqrt 2 {\rm{cis}}\left( { - \dfrac{{7\pi }}{{12}}} \right),{z_3} = \sqrt 2 {\rm{cis}}\left( {\dfrac{{3\pi }}{4}} \right)\)

Question 5
12822

The cube roots of \({\rm{ }} - i{\rm{ }}\) are ?

\(\dfrac{{\sqrt 3 - i}}{2},i,\dfrac{{ - \sqrt 3 - i}}{2}\)

Question 6
12823

The square roots of \({\rm{ }}   1 + i\sqrt 3 {\rm{ }}\)  are ?

\( \pm \dfrac{{\sqrt 2 }}{2}\left( {\sqrt 3 + i} \right)\)

Question 7
12862

The cube roots of \( - 2 + 2i\) are?

\({z_1} = \sqrt 2 {\rm{cis}}\left( {\dfrac{\pi }{4}} \right),{z_2} = \sqrt 2 {\rm{cis}}\left( {\dfrac{{11\pi }}{{12}}} \right),{z_3} = \sqrt 2 {\rm{cis}}\left( { - \dfrac{{5\pi }}{{12}}} \right)\)

Question 8
52082

Solve \(z^6+1=0\) over the complex field

\(\begin{aligned}
&z_1=\cos \left(\frac{\pi}{6}\right)+i \sin \left(\frac{\pi}{6}\right) \\
&z_1=\frac{\sqrt{3}}{2}+\frac{1}{2} i \\
&z_2=i \\
&z_3=-\frac{\sqrt{3}}{2}+\frac{1}{2} i \\
&z_4=-\frac{\sqrt{3}}{2}-\frac{1}{2} i \\
&z_5=-i\\
& z_6=\frac{\sqrt{3}}{2}-\frac{1}{2} i
\end{aligned}\)