Determining the irrationality of a quotient

[trap101]

Determine whether (7)^1/2/(15)^1/3 is either rational or irrational and prove your answer is correct.

So I know that (7)^1/2 is irrational from previous theorems since it is a prime, I also split up (15)^1/3 into (5)^1/3 times (3)^1/3. I previously had shown that (5)^1/3 is also irrational. Doing this question I showed that (3)^1/3 is irrational by the uniqueness of prime factorization.

But my problem lies in showing that this whole quotient is irrational. I don't know where to start. Maybe assume that the quotient is rational and obtain a contradiction? If so how could I start it?

 

[Dick]

Determine whether (7)^1/2/(15)^1/3 is either rational or irrational and prove your answer is correct.

So I know that (7)^1/2 is irrational from previous theorems since it is a prime, I also split up (15)^1/3 into (5)^1/3 times (3)^1/3. I previously had shown that (5)^1/3 is also irrational. Doing this question I showed that (3)^1/3 is irrational by the uniqueness of prime factorization.

But my problem lies in showing that this whole quotient is irrational. I don't know where to start. Maybe assume that the quotient is rational and obtain a contradiction? If so how could I start it?

Yes, assume it's equal to a rational and assume its in lowest terms. Then take the sixth power of both sides. Then make the usual sort of arguments about prime divisibility.