How do you do this? Use a rational exponent to write (exponent 2)√3 * (exponent 4)√5?

so you mean the 2 and 4 are little numbers inside the √ ? if so, they are not exponents but indices. usually you don't' write the two.

If I am correct, then you would write

3^1/2 * 5^1/4

and since you do not have like bases or indices you cannot multiply the terms