let P(x)=0 be a polynomial equation of least possible degree with rational coefficients having $\displaystyle \sqrt[3]7+\sqrt[3]49$ as one of its roots
.then the product of all the roots of P(x)=0 is
You left out an awful lot of information. What kind of course did you see this problem in? What do you know about such problems? What ideas do you have about it? Do you have, for example, a "guess" as to what degree this polynomial should be?
Let $\displaystyle a = 7^{\frac{1}{3}}$. Let $\displaystyle b = a + a^2$.
Note $\displaystyle a^3 = 7$, which will be used to simplify powers of $\displaystyle a$ greater than 2.
$\displaystyle b^3 = (a + a^2)^3 = (a)^3(1+a)^3 = 7(a^3 + 3a^2 + 3a + 1) = 7(7 + 3a^2 + 3a + 1)$
$\displaystyle = 21a^2 + 21a + 56 = 21(a^2 + a) + 56 = 21b + 56$.
Can you complete it from here?