Question

The value of the expression $\left(\frac{{81}^{\frac{1}{{\log }_{5}9}}+3^{\frac{3}{{\log }_{\sqrt{6}}3}}}{409}\right)(7^{\frac{1}{{}^{{\log }_{25}7}}}-5^{\frac{1}{{\log }_{\sqrt{6^3}}5}})$ is

Answer 1

Given,
$\left(\frac{{81}^{\frac{1}{{\log }_{5}9}}+3^{\frac{3}{{\log }_{\sqrt{6}}3}}}{409}\right)(7^{\frac{1}{{}^{{\log }_{25}7}}}-5^{\frac{1}{{\log }_{\sqrt{6^3}}5}})$
$=\left(\frac{{81}^{{\log }_{9}5}+3^{3{\log }_3\sqrt{6}}}{409}\right)(7^{{\log }_{7}25}-5^{{\log }_5{6}^{\frac{3}{2}}})[\because {\log }_{a}b=\frac{1}{{\log }_{b}a}]$
$=\left(\frac{25+6^{\frac{3}{2}}}{409}\right)(25-6^{\frac{3}{2}})[\because m\log a=\log a^m\text{\&}{a}^{{\log }_{a}b}=b]=\frac{625-216}{409}=1$