If $\alpha ,\beta$  be positive real the roots of the equation $x^2-px+q=0$, then the value of ${\alpha }^{1/4}+{\beta }^{1/4}$  is ${\left(p+6\sqrt{q}+4q^{1/4}\sqrt{p+2\sqrt{q}}\right)}^k,$ where k is equal to  
 

Since $\alpha ,\beta$ are the roots of the equation $x^2+px+q=0$

$\therefore \alpha +\beta =panda\beta =q.$

Now, ${({\alpha }^{1/4}+{\beta }^{1/4})}^4={\left[{({\alpha }^{1/4}+{\beta }^{1/4})}^2\right]}^2$

${[{\alpha }^{1/2}+{\beta }^{1/2}+2{\left(\alpha \beta \right)}^{1/4}]}^2={\left[\sqrt{\alpha +\beta +2\sqrt{\alpha \beta }}+2{\left(\alpha \beta \right)}^{1/4}\right]}^2$

$={\left[\sqrt{p+2\sqrt{q}}+2{\left(q\right)}^{1/4}\right]}^2=p+6\sqrt{q}+4q^{1/4}\sqrt{p+2\sqrt{q}}$

$\therefore {\alpha }^{1/4}+{\beta }^{1/4}={\left[p+6\sqrt{q}+4q^{1/4}\sqrt{p+2\sqrt{q}}\right]}^{1/4}$