smc

We look at each statement individually. $\text{I:}2x^x$. We note that $x^x+x^x=x^x(1+1)=2x^x$. So statement $\text{I}$ is true. $\text{II:}{x}^{2x}$. We find a counter example which is $x=1$. $2\ne 1$. So statement $\text{II}$ is false. $\text{III:}(2x{)}^x$. We see that this statement is equal to $2^x{x}^x$. $x=2$ is a counter example. $8\ne 16$. So statement $\text{III}$ is false. $\text{IV:}(2x{)}^{2x}$. We see that $x=1$ is again a counter example. $2\ne 4$. So statement $\text{IV}$ is false. Therefore, our answer is $\boxed{1}$.