Question

Simplification of the expression $\sqrt{\left({\left(\frac{1}{\sqrt{\left(27\right)}}\right)}^{2-\frac{\left({\log }_{5}13\right)}{\left(2{\log }_{5}9\right)}}\right)}$ gives

• A. $3^{-\frac{3}{2}}\times {13}^{\frac{3}{16}}$
• B. $3^{\frac{3}{2}}\times {13}^{\frac{3}{16}}$
• C. $3^{-\frac{3}{2}}\times {13}^{\frac{3}{8}}$
• D. $3^{\frac{3}{2}}\times {13}^{-\frac{3}{16}}$

Answer 1

The correct option is A $3^{-\frac{3}{2}}\times {13}^{\frac{3}{16}}$

Given,
$\sqrt{[{\left(\frac{1}{\sqrt{\left(27\right)}}\right)}^2\times {\left(\frac{1}{\sqrt{\left(27\right)}}\right)}^{-\left({\log }_{5}13\right)/\left(2{\log }_{5}9\right)}]}$

$=\frac{1}{\sqrt{\left(27\right)}}\times {\left(\sqrt{27}\right)}^{\frac{1}{4}{\log }_{9}13}[\because {\log }_{a}b=\frac{\log b}{\log a}]$

$=3^{-3/2}\times {\left(3\right)}^{\frac{3}{8}{\log }_{9}13}=3^{-3/2}\times 3^{\frac{3}{16}{\log }_{3}13}$

$=3^{-3/2}\times 3^{{\log }_3{\left(13\right)}^{3/16}}[\because m\log a=\log a^m]=3^{-3/2}\times {13}^{3/16}[\because a^{{\log }_{a}b}=b]$