If $\alpha$ $\beta$ and $\gamma$ are the roots of the equation ${\mathrm{x}}^3-7\mathrm{x}+7=0\text{, then }\frac{1}{{\mathrm{\alpha }}^4}+\frac{1}{{\mathrm{\beta }}^4}+\frac{1}{{\mathrm{\gamma }}^4}$ is

Here, $\mathrm{\Sigma \alpha }=0,\mathrm{\Sigma \alpha \beta }=-7,\mathrm{\alpha \beta \gamma }=-7$
$\therefore \frac{1}{{\mathrm{\alpha }}^4}+\frac{1}{{\mathrm{\beta }}^4}+\frac{1}{{\mathrm{\gamma }}^4}=\frac{{\mathrm{\alpha }}^4{\mathrm{\beta }}^4+{\mathrm{\beta }}^4{\mathrm{\gamma }}^4+{\mathrm{\gamma }}^4{\mathrm{\alpha }}^4}{{\mathrm{\alpha }}^4{\mathrm{\beta }}^4{\mathrm{\gamma }}^4}$
$=\frac{{\mathrm{\Sigma \alpha }}^4{\mathrm{\beta }}^4}{{\mathrm{\alpha }}^4{\mathrm{\beta }}^4{\mathrm{\gamma }}^4}$…………………………..(i)
Now ,$\mathrm{\Sigma \alpha \beta \Sigma \alpha \beta \Sigma \alpha \beta \Sigma \alpha \beta }=\left(\mathrm{\Sigma \alpha \beta }{)}^2\right(\mathrm{\Sigma \alpha \beta }{)}^2$
$\begin{array}{r}\Rightarrow (-7{)}^4=\left[{\mathrm{\alpha }}^2{\mathrm{\beta }}^2+{\mathrm{\beta }}^2{\mathrm{\gamma }}^2+{\mathrm{\gamma }}^2{\mathrm{\alpha }}^2+2\mathrm{\alpha \beta \gamma }(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })\right]\\ \left[{\mathrm{\alpha }}^2{\mathrm{\beta }}^2+{\mathrm{\beta }}^2{\mathrm{\gamma }}^2+{\mathrm{\gamma }}^2{\mathrm{\alpha }}^2+2\mathrm{\alpha \beta \gamma }(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })\right]\\ =\left({\mathrm{\alpha }}^2{\mathrm{\beta }}^2+{\mathrm{\beta }}^2{\mathrm{\gamma }}^2+{\mathrm{\gamma }}^2{\mathrm{\alpha }}^2\right)\left({\mathrm{\alpha }}^2{\mathrm{\beta }}^2+{\mathrm{\beta }}^2{\mathrm{\gamma }}^2+{\mathrm{\gamma }}^2{\mathrm{\alpha }}^2\right)\\ [\because \mathrm{\Sigma \alpha }=\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }=0]\end{array}$
$\begin{array}{l}={\alpha }^4{\beta }^4+{\beta }^4{\gamma }^4+{\gamma }^4{\alpha }^4+2{\alpha }^4{\beta }^2{\gamma }^2+2{\alpha }^2{\beta }^4{\gamma }^2+2{\alpha }^2{\beta }^2{\gamma }^4\\ =\mathrm{\Sigma }{\alpha }^4{\beta }^4+2{\alpha }^2{\beta }^2{\gamma }^2\left({\alpha }^2+{\beta }^2+{\gamma }^2\right)\\ =\mathrm{\Sigma }{\alpha }^4{\beta }^4+2{\alpha }^2{\beta }^2{\gamma }^2\left[(\mathrm{\Sigma }\alpha {)}^2-2\mathrm{\Sigma }\alpha \beta \right]\\ =\mathrm{\Sigma }{\alpha }^4{\beta }^4+2{\alpha }^2{\beta }^2{\gamma }^2[0-2\times (-7\left)\right]\\ =\mathrm{\Sigma }{\alpha }^4{\beta }^4+2(-7{)}^2(2\times 7)\\ \Rightarrow \mathrm{\Sigma }{\alpha }^4{\beta }^4=(-7{)}^4+4(-7{)}^3\\ \Rightarrow \mathrm{\Sigma }{\alpha }^4{\beta }^4=(-7{)}^3(-7+4)=-3(-7{)}^3\end{array}$
On putting this value in Eq. (i), we get
$\frac{1}{{\mathrm{\alpha }}^4}+\frac{1}{{\mathrm{\beta }}^4}+\frac{1}{{\mathrm{\gamma }}^4}=\frac{-3(-7{)}^3}{(-7{)}^4}=\frac{-3}{-7}=\frac{3}{7}$