Question

The value of ${81}^{\left(\frac{1}{{\log }_{5}3}\right)}+{27}^{{\log }_{9}36}+3^{\frac{4}{{\log }_{7}9}}$ is

Answer 1

$y={81}^{\left(\frac{1}{{\log }_{5}3}\right)}+{27}^{{\log }_{9}36}+3^{\frac{4}{{\log }_{7}9}}$
$\Rightarrow y=3^{4{\log }_{3}5}+{\left(3^3\right)}^{\frac{2{\log }_{3}6}{2}}+3^{2{\log }_{3}7}[\because {\log }_{a^n}{b}^m=\frac{m}{n}{\log }_{a}b\text{\&}{\log }_{a}b=\frac{1}{{\log }_{b}a}]$
$\Rightarrow y=3^{{\log }_3{5}^4}+3^{{\log }_3{6}^3}+3^{{\log }_3{7}^2}=5^4+6^3+7^2[\because a^{{\log }_{a}b}=b]$
Therefore, $y=890$
Ans: 890