$\begin{aligned} \quad \bullet\quad |a|&=|3+i|=\sqrt{3^2+1^2}\\&=\sqrt{9+1}=\sqrt{10} \end{aligned}$

$\begin{aligned} \quad \bullet\quad |b|&=|\sqrt{2}-3i |\\&=\sqrt{\sqrt{2}^2+(-3)^2}=\sqrt{11} \end{aligned}$

$\begin{aligned} \quad \bullet\quad |c|&=\left|\dfrac{1+\sqrt{2}i}{1-3i}\right|=\dfrac{|1+\sqrt{2}i|}{|1-3i|} \\&=\dfrac{\sqrt{1^2+\sqrt{2}^2}}{\sqrt{1^2+(-3)^2}}=\dfrac{\sqrt{30}}{10} \end{aligned}$

$\begin{aligned} \quad \bullet\quad |d| &=|(1+2i)^3|=|1+2i|^3 \\&=\sqrt{1^2+2^2}^3\\&=\sqrt{5}^3=5\sqrt{5} \end{aligned}$