Given that $\sqrt[ ]{3}$ is an irrational number, $5+2\sqrt[ ]{3}$ is not an irrational number.

Suppose $5+2\sqrt[ ]{3}$ is a rational number. 

Therefore, it can be written in $\frac{p}{q}$ form, where p and q are coprime integers. 

 $5+2\sqrt[ ]{3}=\frac{p}{q}$ 

$\Rightarrow \sqrt[ ]{3}=\frac{p-5q}{2q}$ 

Since p and q are integers, therefore $\frac{p-5q}{2q}$ must be a rational number. 

But this is a contradiction as the LHS is an irrational number. 

This suggests that the assumption was wrong. 

Hence, $5+2\sqrt[ ]{3}$ is an irrational number.