What is a number that makes this expression rational?

Question

√2 *x√x*4√10x

Answer 1

Your question makes no sense.
Did you mean:
By what number must (√2)(x√x)(4√(10x)) be multiplied to make it rational ?

(√2)(x√x)(4√(10x))
= 4√20 x^2
= 8√5 x^2

If I multiply this by √5, I get 40x^2

Let me know if that was not your question, in that case, rephrase it so it makes sense.

Answer 2

If you are looking for a number that makes 8√5 x^2 rational, then you need
x^2 = √5
x = ∜5

extra credit -- why not also -∜5 ?