$$ \frac{(xyz)^{\frac{a}{3}}}{abc}=\sqrt{axy}. $$

$$ \frac{(xyz)^{\frac{a}{3}}}{abc} = \sqrt{axy} \quad\Rightarrow\quad \frac{x^{\frac{a}{3}}y^{\frac{a}{3}}z^{\frac{a}{3}}}{abc} = a^{\frac12}x^{\frac12}y^{\frac12}. $$ $$ \Rightarrow \frac{x^{\frac{a}{3}}y^{\frac{a}{3}}z^{\frac{a}{3}}}{abc} – a^{\frac12}x^{\frac12}y^{\frac12} = 0 $$ $$ \Rightarrow abc \left[ \frac{x^{\frac{a}{3}}y^{\frac{a}{3}}z^{\frac{a}{3}}}{abc} – a^{\frac12}x^{\frac12}y^{\frac12} \right] = (abc)(0) $$ $$ \Rightarrow x^{\frac{a}{3}}y^{\frac{a}{3}}z^{\frac{a}{3}} = (abc)a^{\frac12}x^{\frac12}y^{\frac12} $$ $$ \Rightarrow x^{\frac{a}{3}}y^{\frac{a}{3}}z^{\frac{a}{3}} \left(\frac{1}{x^{\frac12}}\right) = (abc)a^{\frac12}x^{\frac12}y^{\frac12} \left(\frac{1}{x^{\frac12}}\right) $$ $$ \Rightarrow x^{\frac{a}{3}-\frac12}(yz)^{\frac{a}{3}} = a^{1+\frac12}bc\sqrt{y} = a^{\frac32}bc\sqrt{y} $$ $$ \Rightarrow x^{\frac{a}{3}-\frac12} = a^{\frac32}bc\sqrt{y}(yz)^{-\frac{a}{3}} $$ $$ \Rightarrow x^{\frac{a}{3}-\frac12} = a^{\frac32}bcy^{\frac12}y^{-\frac{a}{3}}z^{-\frac{a}{3}} $$ $$ \Rightarrow x^{\frac{2a-3}{6}} = a^{\frac32}bcy^{\frac12-\frac{a}{3}}z^{-\frac{a}{3}} = \frac{a^{\frac32}bcy^{\frac{3-2a}{6}}}{z^{\frac{a}{3}}} $$ $$ \Rightarrow x^{\frac{2a-3}{6}} = \frac{a^{\frac32}bcy^{\frac{3-2a}{6}}}{z^{\frac{a}{3}}} \quad\Rightarrow\quad x = \left( \frac{a^{\frac32}bcy^{\frac{3-2a}{6}}}{z^{\frac{a}{3}}} \right)^{\frac{6}{2a-3}}. $$