3. In the set of complex numbers, consider the equations $x^{19}=-i$ and $y^{53}=i$

a) Calculate $(a \cdot b)^{2014}$ where $a$ is a solution to one of the equations and $b$ is a solution to the other equation.

b) Find the common solutions of the two equations.

Prof. Voiculeț Septimius, Videle
3. a) We can consider a solution to the first equation.

$(a \cdot b)^{2014}=a^{2014} \cdot b^{2014}=\left(a^{19}\right)^{106} \cdot\left(b^{53}\right)^{38}=(-i)^{106} \cdot i^{38}=(-1)(-1)=1$

b) If $z$ is a common solution, $z^{19}=-i, z^{53}=i,\left(z^{19}\right)^{3}=(-i)^{3}=i$

$3 p$

$$
\begin{aligned}
& \text { so } z^{57}=i \text { and } z^{53}=i, \text { it follows that } z^{4}=1 \text {, i.e., } z \in\{1,-1, i,-i\} \ldots \ldots \ldots \ldots . \ldots 2 \\
& z=1, z=-1 \text { and } z=-i \text { do not satisfy both equations, while } i \text { does. } \\
& \text { Therefore, } z=i \text { is the only common solution...................................... }
\end{aligned}
$$