$$7+3\sqrt[ ]{2}$$ is not a rational number. 

f possible, suppose $7+3\sqrt[ ]{2}$ to be rational. 

Therefore, we can find two integers a and b (b ≠ 0) such that 

$$7+3\sqrt[ ]{2}=\frac{a}{b}$$ 

$$\Rightarrow \sqrt[ ]{2}=\frac{1}{3}\left(\frac{a}{b}-7\right)$$ 

Since a and b are rational, $\frac{1}{3}\left(\frac{a}{b}-7\right)$ should also be rational and thus, $\sqrt[ ]{2}$ is rational as well. This contradicts the fact that $\sqrt[ ]{2}$ is irrational. 

Therefore, our assumption is false and hence, $7+3\sqrt[ ]{2}$ is not a rational number.