Writing this radical using rational exponents

(5 4√10x)^3

The 4 is tiny inside the square root!

This is what I did so far
( 5 (10^ 1/4 x ^1/4) ^3
125 10^3/4 x^3/4

I don't know if I'm doing it right.

if by saying "4 is the tiny inside the square root!" you mean you are taking the 4th root of 10x then you are doing it wrong

first (5 4sqrt(10x))^3=(5 4sqrt(10x))*(5 4sqrt(10x))*(5 4sqrt(10x))=125*(10x^3/4)

yes you are right. Final expression is [2^(3/4)]*[5^(15/4)]*[x(3/4)]

Correct.