Mathematics - Binomial Theorem Question with Solution | TestHub

${\left\{3^{{\log }_3\sqrt{{25}^{x-1}+7}}+3^{\left(-\frac{1}{8}\right){\log }_3\left(5^{x-1}+1\right)}\right\}}^{10}$

$={\left\{\sqrt{{25}^{x-1}+7}+{\left(5^{x-1}+1\right)}^{\frac{-1}{8}}\right\}}^{10},$ using $a^{n{\log }_{a}x}=x^n$

Using the formula of the general term in the Binomial theorem for non negative integral index, i.e., $T_{r+1}={}^{n}C_r{a}^{n-r}{b}^r$ in ${\left(a+b\right)}^n,$ we get,  $T_9=T_{8+1}={}^{10}C_8\left({25}^{\left(x-1\right)}+7\right)\times {\left(5^{\left(x-1\right)}+1\right)}^{-1}=180$

$\Rightarrow \frac{{25}^{x-1}+7}{5^{\left(x-1\right)}+1}=4$

$\Rightarrow \frac{t^2+7}{t+1}=4,$ taking $t=5^{x-1}$

$\Rightarrow t^2+7=4\left(t+1\right)$

$\Rightarrow t^2-4t+3=0$

$\Rightarrow \left(t-3\right)\left(t-1\right)=0$

$\Rightarrow t=1,3\Rightarrow 5^{x-1}=1\Rightarrow 5^{x-1}=5^0$

$\Rightarrow x-1=0$ (one of the possible value).

$\Rightarrow x=1$