When 3^4√2 written as an entire radical 4√n= ?

3^4√2

81√2
4√(81^2/16 * 2)
4√(6561/8)

somehow I doubt that's what you were after

if by 4√n you mean 4th root (∜n) I'm also confused.

3^∜2 is trés bizarre.

Try again using ^ for exponents and √ for square root, and ∜ for 4th root.