Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 2 Exercises 2.5
Our main tool is Corollary 2.3.
Solution: First, we need to find an equation with integer coefficients such that $(3+\sqrt{2})^{2/3}$ is a solution to it.
Let $x=(3+\sqrt{2})^{2/3}$. Taking cube for both sides, we obtain that $x^3=(3+\sqrt{2})^{2}$. Hence
$$
x^3=9+6\sqrt 2+2=11+6\sqrt 2.
$$ In particular, $x^3-11=6\sqrt 2$. Squaring both sides, we get
$$
(x^3-11)^2=72,
$$ which is
$$
x^6-22x^3+49=0.
$$ Consider the rational solution of $x^6-22x^3+49=0$. It follows from Corollary 2.3 that those rationa solutions can only be $\pm 1$,$\pm 7$, $\pm 49$.
Direct computations would show that none of them will be a solution of $x^6-22x^3+49=0$. Hence $x^6-22x^3+49=0$ has no rational solutions. Therefore, as a solution to $x^6-22x^3+49=0$, $(3+\sqrt{2})^{2/3}$ is irrational.