Question

Encontre as raízes quadradas de z= 4+4 √3i

Answer 1

Resposta:

$\sqrt{6}+\sqrt{2}i~~ ou ~~\sqrt{6}-\sqrt{2}i$

Explicação passo a passo:

a + bi a raiz de Z

$\sqrt{4+4\sqrt{3}i } =a+bi\\\\4+4\sqrt{3}i=(a+bi)^2 \\\\4+4\sqrt{3}i =a^2+2abi+b^2i^2\\\\a^2-b^2+2abi=4+4\sqrt{3}i \\\\\left \{ {{a^2-b^2=4} \atop {2ab=4\sqrt{3} }} \right. \\\\b=\frac{4\sqrt{3} }{2a} =\frac{2\sqrt{3} }{a} \\\\a^2-(\frac{2\sqrt{3} }{a})^2 =4\\\\a^4-\frac{4.3}{a^2} =4\\\\a^4-4a^2-12=0$

$\Delta=(-4)^2-4.1.(-12)\\\\\Delta=16+48\\\\\Delta=64\\\\a^2=\frac{-(-4)\pm\sqrt{64} }{2.1}\\\\ a^2=\frac{4\pm8}{2}\\\\a=\pm\sqrt{6} ~~ ou~~a^2=-2 (nao~~serve)\\\\b=\frac{2\sqrt{3} }{\sqrt{6} } \\\\b=\frac{2\sqrt{3} }{\sqrt{2}.\sqrt{3} } \\\\b=\frac{2}{\sqrt{2} } =\frac{2\sqrt{2} }{2}=\sqrt{2} \\\\a+bi=\sqrt{6}+\sqrt{2}i\\\\b=\frac{2\sqrt{3} }{-\sqrt{6} } \\\\b=\frac{2\sqrt{3} }{-\sqrt{2}\sqrt{3} } =-\frac{2}{\sqrt{2} }=-\frac{2\sqrt{2} }{2} =-\sqrt{2}\\\\a+bi=-\sqrt{6}-\sqrt{2}i$