Proving Multiplicative Group of Positive Rationals Generated by Primes

[futurebird]

From "The Theory of Groups" by Rotman

2.5. Prove that the multiplicative group of positive rationals is generated by all rationals of the form:
$$\frac{1}{p}$$,
where p is prime.​

... um... no it's not. Right? How can I prove this when I don't even think it is true? I mean, for example take the positive rational number 75. How can I generate that using $$\frac{1}{p^{n}}$$, where p is prime, $$n \in \mathbb{N}$$? The largest possible value for $$\frac{1}{p^{n}}$$ is 1/2...

I could see how this would be possible if we had addition as the operation for the generating set... but, then why is it the multiplicative group of positive rationals?

Please help me understand this question.

 

[Hurkyl]

for example take the positive rational number 75. How can I generate that using $$\frac{1}{p^{n}}$$, where p is prime, $$n \in \mathbb{N}$$?

You can't, but why would you restrict yourself to positive exponents?

 

[futurebird]

You can't, but why would you restrict yourself to positive exponents?

Thanks!