BMSAT Kozhikode: AlgebraIdentities. Free, no login required.

If $x = \sqrt[3]{28}$ and $y = \sqrt[3]{27}$, find the value of $x + y - \frac{1}{x^2+xy+y^2}$.

Say $y = 3\sqrt{2} - 4$, find $y^2 + \frac{1}{y^2}$.

If $x^{2}+\frac{1}{x^{2}}=83$ and $x>1$, then the value of $x^{3}-\frac{1}{x^{3}}$ is