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

Let us assume, to the contrary that  $5+3\sqrt[ ]{2}$  is rational.

That is, we can find coprime a and b (b$\ne$0) such that  $5+3\sqrt[ ]{2}=\frac{a}{b}$ 

Therefore,

 $5-\frac{\mathrm{a}}{\mathrm{b}}=3\sqrt{2}$ 

$\Rightarrow \sqrt[ ]{2}=\frac{5\mathrm{b}-\mathrm{a}}{3\mathrm{b}}$ 

Since, a and b are integers, we get $\frac{5b-a}{3b}$ is rational, and so $\sqrt[ ]{2}$ is rational which contradicts the fact that  $\sqrt[ ]{2}$ is irrational. 

So, our assumption was wrong that  $5+3\sqrt[ ]{2}$  is rational. 

Hence, we conclude that  $5+3\sqrt[ ]{2}$  is irrational.