The given expression is
[tex]\frac{\sqrt{54a^3}}{\sqrt{2a}}[/tex]
Apply radical rule:
[tex]\begin{gathered} \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}},\:\quad \mathrm{\:assuming\:}a\ge 0,\:b\ge 0 \\ \frac{\sqrt{54a^3}}{\sqrt{2a}}=\sqrt{\frac{54a^3}{2a}} \end{gathered}[/tex]
Cancel the common factor (2a), we will have
[tex]=\sqrt{27a^2}[/tex]
Apply radical rule:
[tex]\begin{gathered} \sqrt{ab}=\sqrt{a}\sqrt{b},\:\quad \mathrm{\:assuming\:}a\ge 0,\:b\ge 0 \\ \sqrt{27a^2}=\sqrt{27}\sqrt{a^2} \\ \sqrt{a^2}=a,\:\quad \mathrm{\:assuming\:}a\ge 0 \\ =\sqrt{27}a \end{gathered}[/tex]
Then
[tex]\sqrt{27}a=3\sqrt{3}a=3a\sqrt{3}[/tex]
Hence, the answer is
[tex]3a\sqrt{3}\text{ \lparen Option 2\rparen}[/tex]
