[tex]a = 42^{15} = 42^{3\cdot 5}[/tex]
[tex]b = 35^{18} = 35^{2\cdot 3\cdot 3}[/tex]
[tex]42^{3\cdot 5} \,\square \, 35^{2\cdot 3\cdot 3}\Big|^{\frac{1}{3}} \Leftrightarrow 42^5 \,\square\,35^{2\cdot 3}\Leftrightarrow 42^5 \,\square\,35^{5}\cdot 35\Big|^{\frac{1}{5}} \Leftrightarrow[/tex]
[tex]\Leftrightarrow 42 \,\square\,35\cdot \sqrt[5]{35}[/tex]
[tex]2^5 < 35 \Rightarrow 2 < \sqrt[5]{35} \Rightarrow 35\cdot 2 < 35\cdot \sqrt[5]{35}[/tex]
[tex]\Rightarrow 42 <(35\cdot 2 <35\cdot \sqrt[5]{35}) \Leftrightarrow 42 \,\boxed{<} \,35\cdot \sqrt[5]{35}[/tex]
[tex]\Rightarrow \boxed{42^{15} < 35^{18}}[/tex]
