Question

The value of ${(3^{\sqrt{{\log }_{3}4}}-4^{\sqrt{{\log }_{4}3}})}^2$ is

Answer 1

${(3^{\sqrt{{\log }_{3}4}}-4^{\sqrt{{\log }_{4}3}})}^2$
Let ${\log }_{3}4=t$
$\Rightarrow {\log }_{4}3=\frac{1}{t}$ and $3^t=4$

$\therefore 3^{\sqrt{{\log }_{3}4}}-4^{\sqrt{{\log }_{4}3}}$
$=3^{\sqrt{t}}-(3^t{)}^{\frac{1}{\sqrt{t}}}$
$=3^{\sqrt{t}}-3^{\sqrt{t}}=0$