Question

Me pueden ayudar con este ejercicio.

Answer 1

1)
$|z|=\left|\left(\dfrac{1}{\sqrt{2}}-\dfrac{1}{\sqrt{2}}i\right)^3(3-4i)\right|\\ \\ \\ |z|=\left|\left(\dfrac{1}{\sqrt{2}}-\dfrac{1}{\sqrt{2}}i\right)^3\right|\left|(3-4i)\right|\\ \\ \\ |z|=\left|\left(\dfrac{1}{\sqrt{2}}-\dfrac{1}{\sqrt{2}}i\right)\right|^3\left|(3-4i)\right|\\ \\ \\ |z|=\sqrt{\left( \dfrac{1}{\sqrt{2}}\right)^2+\left( \dfrac{1}{\sqrt{2}}\right)^2}^3\times \sqrt{3^2+4^2}\\ \\ \\ |z|=5$

2)
$|w|=\left|\dfrac{\sqrt{\sqrt{15}+i}~(3+5i)}{\sqrt{34}i}\right|\\ \\ \\ |w|=\dfrac{\left|\sqrt{\sqrt{15}+i}\right|~\left|(3+5i)\right|}{\left|\sqrt{34}i\right|}\\ \\ \\ |w|=\dfrac{\left|\sqrt{\sqrt{15}+i}\right|~\sqrt{34}}{\sqrt{34}}\\ \\ \\ |w|=\left|\sqrt{\sqrt{15}+i}\right|\\ \\ \\ |w|=\sqrt{\left|\sqrt{15}+i\right|}\\ \\ \\ |w|=4$