Powers and roots of complex numbers

1103109303

Level: 
B
Consider an equation \( x^n+b=0 \), where \( n \) is a positive integer and \( b \) is a real number. The points that correspond to the roots of the equation are marked in the figure as black points. Find the equation.
\( x^8 - 256 = 0 \)
\( x^8 + 256 = 0 \)
\( x^4 + 16 = 0 \)
\( x^4 - 16 = 0 \)
\( x^6 - 64 = 0 \)
\( x^6 + 64 = 0 \)

2000002602

Level: 
B
Consider the equation \(x^4 =1\), where \(x\) is a complex variable. Which of the following statements is true?
The equation has four different complex roots.
The equation has no real root.
The equation has two double roots: \(x_{1,2}=1\) and \(x_{3,4}=-1\).
The equation has the root \(x=1+i\).

2000002604

Level: 
B
Find the solution set of the equation \(x^4+81=0\) if you know that one of its roots is \(\frac{3}{\sqrt{2}}(1+i)\).
\( \left\{ \frac{3}{\sqrt{2}}(1+i); -\frac{3}{\sqrt{2}}(1+i); \frac{3}{\sqrt{2}}(1-i);-\frac{3}{\sqrt{2}}(1-i) \right\} \)
\( \left\{ \frac{3}{\sqrt{2}}(1+i); -\frac{3}{\sqrt{2}}(1+i);3;-3 \right\} \)
\( \left\{ \frac{3}{\sqrt{2}}(1+i); \frac{3}{\sqrt{2}}(1-i);3i;-3i \right\} \)
\( \left\{\frac{3}{\sqrt{2}}(1+i);\frac{3}{\sqrt{2}}(1-i) \right\}\)

2000002606

Level: 
B
Imagine all the solutions of the equation \(x^6 -64 =0\) shown as points in the complex plane. Find the false statement.
Two points lie on the imaginary axis.
The values of the arguments of any two solutions differ by an integer multiple of \(\frac{\pi}{3}\).
All solutions of the equation lie on a circle centered at the origin with a radius of \(2\).
Two points lie on the real axis.

2000002608

Level: 
B
Find the right formula for solving the equation \(x^5 +32=0\).
\( x_k = \sqrt[5]{|-32|}( \cos\frac{\pi +2k\pi}{5}+ i\sin \frac{\pi +2k\pi}{5})\), \(k=0,1,2,3,4\)
\( x_k = \sqrt[5]{-32}( \cos\frac{\pi +2k\pi}{5}+ i\sin \frac{\pi +2k\pi}{5})\), \(k=0,1,2,3,4\)
\( x_k = \sqrt[5]{|-32|}( \cos \frac{\pi +k\pi}{5}+ i\sin \frac{\pi +k\pi}{5})\), \(k=0,1,2,3,4\)
\( x_k = \sqrt[5]{|-32|}( \cos \frac{\pi +2k\pi}{5}+ \sin \frac{\pi +2k\pi}{5})\), \(k=0,1,2,3,4\)