To simplify the expression \((3^6/3^{-4})^2\), we can first simplify the fraction inside the parentheses:
\[ \frac{3^6}{3^{-4}} = 3^{6 - (-4)} = 3^{6 + 4} = 3^{10} \]
Next, we raise \(3^{10}\) to the power of 2:
\[ (3^{10})^2 = 3^{10 \times 2} = 3^{20} \]
So the expression \((3^6/3^{-4})^2\) simplifies to \(3^{20}\).
All expressions that are equivalent to \(3^{20}\) will include:
- \(3^{20}\)
- \(81^{10}\) (since \(81 = 3^4\) and \(81^{10} = (3^4)^{10} = 3^{4 \times 10} = 3^{40}\))
- \(9^{10}\) (since \(9 = 3^2\) and \(9^{10} = (3^2)^{10} = 3^{2 \times 10} = 3^{20}\))
- \(27^{\frac{20}{3}}\) (since \(27 = 3^3\) then \(27^{\frac{20}{3}} = (3^3)^{\frac{20}{3}} = 3^{3 \times \frac{20}{3}} = 3^{20}\))
Thus, the equivalent expressions are those that simplify to \(3^{20}\).