Question

Решить комплексные числа
1.z^2-3i=3;
2.5z^3+625i=0;

Answer 1

$1. z=\sqrt[4]{18} *e^{i\frac{\pi }{8} }, z=-\sqrt[4]{18} *e^{i\frac{\pi }{8} }\\2. z=5i$

Пошаговое объяснение:

1.

$z^{2} -3i=3 = z^{2}=3+3i\\z_{2} = x_{2}+ iy_{2}=3+3i \\= r_{2}=\sqrt{x_{2}^{2} + y_{2}^{2} } =\sqrt{9+9} =\sqrt{18}\\x_{2}0, y_{2}0 = \alpha _{2}=arctg|\frac{y_{2}^{2}}{x_{2}^{2}} |=arctg(1)=\frac{\pi }{4} \\= z_{2} =r_{2}*e^{i\alpha_{2} } =\sqrt{18}*e^{i\frac{\pi }{4} } \\= z=+-\sqrt{z_{2}} =+-\sqrt[4]{18} *e^{i\frac{\pi }{8} }$

2.

$5z^{3} +625i=0\\5*(z^{3} +125i)=0\\z^{3} = -125i\\z^{3} = 5^{3}*i^{2} * i = 5^{3}*i^{3} \\<img src=$" alt="1. z=\sqrt[4]{18} *e^{i\frac{\pi }{8} } , -\sqrt[4]{18} *e^{i\frac{\pi }{8} }\\2. z=5i" />" />