Writing a number as a linear combination using only coefficients in $\mathbb{Q}$ Can someone please tell me how to write $\frac{1}{7+\sqrt[3]{2}}$
as a linear combination of $1$, $\sqrt[3]{2}$ and $\sqrt[3]{2}^2$using
only coefficients in $\mathbb{Q}$ ?

Let $\omega = \sqrt[3]{2}$, it has minimal polynomial $\omega^3 - 2$.
Like when you divide complex numbers ($\frac{1}{a + ib} = \frac{a - ib}{a^2 + b^2}$) an important part of this is the norm (recall $N(a + ib) = a^2 + b^2$) but unlike the complex case, the automorphisms of the field don't help (because there are not automorphisms of $\mathbb Q(\omega)$!).
The definition of the norm in the general (non-Galois) case is not by multiplying all the conjugates together but the determinant of the linear operator "multiplication by $a + \omega b + \omega^2 c$". So write [a,b,c] as the vector representation of that number. Multiplication by it is represented by the matrix [a,b,c; 2c,a,b; 2b,2c,a] (not that [1,0,0]*that = [a,b,c]) or $$\left( \begin{array}{ccc}
a & b & c \\
2c & a & b \\
2b & 2c & a \end{array} \right)$$ so we have $N(a + \omega b + \omega^2 c) = \text{det}(\text{that matrix}) = a^3 - 6abc + 2b^3 + 4c^3.$ So dividing by this number is represented by the inverse of that matrix, and we know $\text{det}(M)M^{-1}$ has a nice simple form so let's compute [1,0,0]*det(m)/m = [a^2 - 2*c*b, -b*a + 2*c^2, -c*a + b^2] to find $N(a + \omega b + \omega^2 c)/(a + \omega b + \omega^2 c) = (a^2-2bc) + (-ab + 2c)\omega + (-ac + b^2)\omega^2.$
Putting $a = 7$, $b = 1$, $c = 0$ this gives $$\frac{1}{7 + \omega} = \frac{49 - 7 \omega + \omega^2}{345}.$$
