The first step is correct, but the second step is not.
The rule to use is [tex]a^b*a^c = a^{b+c}[/tex]
Emily mistakenly multiplied the exponents when she should have added them.
This is what her steps should look like
[tex]3^{2/3}*\sqrt[6]{3^3}\\\\ = 3^{2/3}*3^{3/6} \\\\= 3^{2/3+3/6} \\\\= 3^{4/6+3/6} \\\\= 3^{7/6} \\\\= \sqrt[6]{3^7} \\\\= \sqrt[6]{2187}[/tex]
Therefore,
[tex]3^{2/3}*\sqrt[6]{3^3}= \sqrt[6]{2187}[/tex]
I used WolframAlpha to confirm the answer.
