z=a+bi
|z^(6)|=|z|^(6)
[tex] |z|^{6} = ({6 \sqrt{2} })^{6} = {6}^{6} \times 8[/tex]
|z|=8^(3)=512
[tex]sqrt{ \sqrt{4 - 3 \sqrt{2}}^{2} + \sqrt{4 + 3 \sqrt{2} } ^{2} } =sqrt{4- 3 \sqrt{2} + 4 + 3 \sqrt{2} } = sqrt{8} [/tex]
[tex]\left|\left(\sqrt{4-3\sqrt{2}}-i\sqrt{4+3\sqrt{2}}\right)^6\right| = \\ \\ = \left|\sqrt{4-3\sqrt{2}}-i\sqrt{4+3\sqrt{2}}\right|^6=\\ \\ = \left[\sqrt{\left(\sqrt{4-3\sqrt{2}}\right)^2+\left(\sqrt{4+3\sqrt{2}}\right)^2}\right]^6=\\ \\ = \left(\sqrt{4-3\sqrt{2}+4+3\sqrt{2}}\right)^6=\\ \\ = \left(\sqrt{8}\right)^6 = 8^3 = \boxed{512}[/tex]
