Let  n be an  even  positive  integer  such that $\frac{n}{2}$  is odd  and let  ${\alpha }_0,{\alpha }_1,\mathrm{.........}{\alpha }_{n-1}$  be the  
complex  roots  of  unity of  order n .When  $\prod _{k=0}^{n-1}(a+b{\alpha }_k^2)={(a^{\frac{n}{t}}+b^{\frac{n}{p}})}^q$ for  any complex numbers  a and b  then  $t+p+q=$ …..

Let the complex number  $\alpha$  such that  ${\alpha }^2=\frac{a}{b}$
 $f\left(x\right)=x^n-1=(x-{\alpha }_0)(x-{\alpha }_1)(x-{\alpha }_2)\mathrm{........}(x-{\alpha }_{n-1})$
We  have  $f\left(\frac{\alpha }{i}\right)={\left(\frac{1}{i}\right)}^n.(\alpha -i{\alpha }_0)(\alpha -i{\alpha }_1)\mathrm{.......}(\alpha -i{\alpha }_{n-1})$
 $f\left(\frac{\alpha }{i}\right)=f{(-\frac{1}{i})}^n.(\alpha +i{\alpha }_0)(\alpha +i{\alpha }_1)\mathrm{.......}(\alpha +i{\alpha }_{n-1})$
Hence ,
 $$\begin{gathered} f\left(\frac{\alpha }{i}\right)f(-\frac{\alpha }{i})=({\alpha }^2+a_0^2)({\alpha }^2-{\alpha }_1^2)\mathrm{........}({\alpha }^2-a_{n-1}^2) \\ =b^n\prod _{k=0}^{n-1}(\alpha +{\alpha }_k^2) \\ {b}^{n}f\left(\frac{\alpha }{i}\right).f(-\frac{\alpha }{i})=b^n.{[{\left(a^2\right)}^{2m+1}+1]}^2 \\ =b^n{[{\left(\frac{a}{b}\right)}^{2m+1}+1]}^2 \\ {b}^{2(2m+1)}{\left(\frac{a^{2m+1}+b^{2m+1}}{b^{2m+1}}\right)}^2 \\ =(a^{n/2}+b^{n/2}) \end{gathered}$$