Question

Let $\alpha, \beta, \gamma$ and $\delta$ be the roots of equation $x^4-3 x^3+5 x^2-7 x+9=0$. 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 $\left(a^b+b^a+a^a+b^b+a b\right)$.

Answer 1

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 $\left(a^b+b^a+a^a+b^b+a b\right)=4^5+5^4+4^4+5^5+4.5=1024+625+256+3125+20=5050$