Proofs involving sets for $\{9^n: n∈ℚ\}=\{3^n: n∈ℚ\}$.

Question

I need to prove that {9^n: n∈ℚ)={3^n: n∈ℚ).

So far I have proven {9^n: n∈ℚ}⊆{3^n: n∈ℚ}.

a∈{9^n: n∈ℚ}. Meaning for some a=9^n for some rational number. Thus, a=9^n=3^2n, showing that a is a rational number for some power of 3. so a∈{9^n: n∈ℚ}. Which also means {9^n: n∈ℚ}⊆{3^n: n∈ℚ}

But i don't know how to prove {3^n: n∈ℚ}⊆{9^n: n∈ℚ}.

So that I can say that {9^n: n∈ℚ)={3^n: n∈ℚ).

Answer 1

Hint:$$\mathbb{Q}=\frac{1}{2}\mathbb{Q}$$

So $9^\mathbb{Q}=9^{\frac{1}{2}\mathbb{Q}}=(9^\frac{1}{2})^\mathbb{Q}=3^\mathbb{Q}$.