1. $\log_a\left(\dfrac{a^2\sqrt[3]{a^4}}{\sqrt[4]{a}}\right)$
$=\log_a\left(\dfrac{a^2a^{\frac{4}{3}}}{a^{\frac{1}{4}}}\right)$
$=\log_aa^{\frac{37}{12}}=\dfrac{37}{12}$
2. $\dfrac{a^{2\sqrt2}-b^{2\sqrt3}}{(a^{\sqrt2}-b^{\sqrt3})^2}+1$
$=\dfrac{(a^{\sqrt2}-b^{\sqrt3})(a^{\sqrt2}+b^{\sqrt3})}{(a^{\sqrt2}-b^{\sqrt3})^2}+1$
$=\dfrac{a^{\sqrt2}+b^{\sqrt3}}{a^{\sqrt2}-b^{\sqrt3}}+1$
$=\dfrac{2a^{\sqrt2}}{a^{\sqrt2}-b^{\sqrt3}}$