$\begin{align} & \text{Express the following in terms of }\alpha \beta \text{ and }\alpha +\beta \\ & (i){{\alpha }^{2}}-{{\beta }^{2}} \\ & (ii){{\alpha }^{2}}+{{\beta }^{2}} \\ & (iii){{\alpha }^{3}}-{{\beta }^{3}} \\ & (iv){{\alpha }^{3}}+{{\beta }^{3}} \\ & (v){{\alpha }^{4}}-{{\beta }^{4}} \\ & (vi)\frac{1}{\alpha }+\frac{1}{\beta } \\ & (vii)\frac{1}{{{\alpha }^{2}}}+\frac{1}{{{\beta }^{2}}} \\\end{align}$
$\begin{align} & (i){{\alpha }^{2}}-{{\beta }^{2}}=(\alpha +\beta )(\alpha -\beta )=(\alpha +\beta )\sqrt{{{(\alpha +\beta )}^{2}}-4\alpha \beta } \\ & (ii){{\alpha }^{2}}+{{\beta }^{2}}={{(\alpha +\beta )}^{2}}-2\alpha \beta \\ & (iii){{\alpha }^{3}}-{{\beta }^{3}}=\left( \sqrt{{{(\alpha +\beta )}^{2}}-4\alpha \beta } \right)\left[ {{\left( \alpha +\beta \right)}^{2}}-\alpha \beta \right] \\ & (iv){{\alpha }^{3}}+{{\beta }^{3}}={{(\alpha +\beta )}^{3}}-3\alpha \beta (\alpha +\beta )=(\alpha +\beta )\left[ {{(\alpha +\beta )}^{2}}-3\alpha \beta \right] \\ & (v){{\alpha }^{4}}-{{\beta }^{4}}=({{\alpha }^{2}}-{{\beta }^{2}})({{\alpha }^{2}}+{{\beta }^{2}})=\left[ (\alpha +\beta )\sqrt{{{(\alpha +\beta )}^{2}}-4\alpha \beta } \right]\left[ {{(\alpha +\beta )}^{2}}-2\alpha \beta \right] \\ & (vi)\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta } \\ & (vii)\frac{1}{{{\alpha }^{2}}}+\frac{1}{{{\beta }^{2}}}=\frac{{{\alpha }^{2}}+{{\beta }^{2}}}{{{(\alpha \beta )}^{2}}}=\frac{{{(\alpha +\beta )}^{2}}-2\alpha \beta }{{{(\alpha \beta )}^{2}}} \\\end{align}$