Here's another approach...

Given: \(3^{6x-3} = \frac{2}{3}\)

For the left side, use the Quotient rule in reverse to get: \(\frac{3^{6x}}{3^{3}} = \frac{2}{3}\)

Evaluate then denominator to get: \(\frac{3^{6x}}{27} = \frac{2}{3}\)

Multiply both sides of the equation by \(27\) to get: \(3^{6x} = 18\)

Raise both sides to the power of \(\frac{1}{2}\) to get: \((3^{6x})^{\frac{1}{2}} = 18^{\frac{1}{2}}\)

Simplify: \(3^{3x} = \sqrt{18} = (\sqrt{9})(\sqrt{2}) = 3\sqrt{2}\)

Answer: E