Question

Let $\alpha ,\beta ,\gamma$ and $\delta$ be the roots of equation $x^4-3x^3+5x^2-7x+9=0$. If the value of $\left|\tan \left({\tan }^{-1}\alpha +{\tan }^{-1}\beta +{\tan }^{-1}\gamma +{\tan }^{-1}\delta \right)\right|=\frac{a}{b}$ where a and $b$ are coprime to each other, then find the value of $\left(a^b+b^a+a^a+b^b+ab\right)$.

Answer 1

Answer: 5050

From given equation, we have
$S_1=\Sigma \alpha =3,S_2=\Sigma \alpha \beta =5$
$S_3=\Sigma \alpha \beta \gamma =7\text{ and }{S}_4=\alpha \beta \gamma \delta =9$
Let ${\tan }^{-1}\alpha =A,{\tan }^{-1}\beta =B,{\tan }^{-1}\gamma =C\&{\tan }^{-1}\delta =D$
Now $\left|\tan \left({\tan }^{-1}\alpha +{\tan }^{-1}\beta +{\tan }^{-1}\gamma +{\tan }^{-1}\delta \right)\right|$
$=|\tan (A+B+C+D)|=\left|\frac{S_1-S_3}{1-S_2+S_4}\right|=\left|\frac{3-7}{1-5+9}\right|=\frac{4}{5}=\frac{a}{b}$
Hence $a=4$ and $b=5$
So $\left(a^b+b^a+a^a+b^b+ab\right)=4^5+5^4+4^4+5^5+4.5=1024+625+256+3125+20=5050$