Is powers of rationals dense in $\mathbb R$ Consider $\mathbb {\tilde {Q}} = \{ x^n : x \in \mathbb Q \} $ $n$ is fixed odd integer.
I have two questions here.


*

*Is this set dense in $\mathbb R$ and

*Is there any bijection exists between $\mathbb Q$ and $\mathbb {\tilde Q}$


For the first question, I think the set is dense. Consider $ a,b \in \mathbb R$ WLOG assume $ a,b \ge 0$ and $ b \ge a$. We can find $c \in \mathbb Q$ such that $ a^{1/n} \le c \le b^{1/n}$. And now $ c^n \in \mathbb {\tilde Q}$ and $ a \le c^n \le b$.
So the set is dense. 
Is there a way to define bijection between $\mathbb Q$ and $\mathbb {\tilde Q}$

Here is a simpler proof of density. Consider the map $f(x)=x^n$, $n>0$ is odd, $f: {\mathbb R}\to {\mathbb R}$. This map is clearly continuous. The intermediate value theorem implies that this map is surjective. The set of rational numbers is dense in ${\mathbb R}$. Therefore, its image under the continuous map $f$ is also dense in $f({\mathbb R})={\mathbb R}$.  
