Multiplication of real and complex radicals If I have, for example, the product

$\sqrt{7+\sqrt{22}}\sqrt[3]{38+i\sqrt{6}} $

Can I perform the multiplication or this cannot be done and only remains to leave the product in this form?
Your question is ambiguous, because there is no such thing as a radical for complex numbers. It's not well defined, because you can't choose a specific complex number for your function (on $\mathbb R$ we choose the positive one).

So if you define $\sqrt.$ properly, may be you could perform your product, but it seems a lot of trouble.
$\sqrt{7+\sqrt{22}}\sqrt[3]{38+\sqrt6i}=
\sqrt[6]{(7+\sqrt{22})^3}\sqrt[6]{(38+\sqrt6i)^2}=
\sqrt[6]{(7+\sqrt{22})^3(38+\sqrt6i)^2}=
\sqrt[6]{1157590+243022\sqrt{22}+(61180\sqrt6+25688\sqrt{33})i}$    

EDIT:
Other possible valid results are the above result multiplied by $\left(\cos{\frac{k\pi}3}+i\sin{\frac{k\pi}3}\right)\text{, where}\;k=\{1\dots 5\}$