To evaluate the expression \( 3^{\frac{2}{3} - 2} \), we first need to simplify it:
- Rewrite \( 2 \) as a fraction with the same denominator: \[ 2 = \frac{6}{3} \] Thus, we can rewrite the expression as: \[ 3^{\frac{2}{3} - \frac{6}{3}} = 3^{\frac{2 - 6}{3}} = 3^{\frac{-4}{3}} \]
Now, we can analyze this result \( 3^{\frac{-4}{3}} \):
- According to the laws of exponents: \[ 3^{\frac{-4}{3}} = \frac{1}{3^{\frac{4}{3}}} \] This can also be rewritten as: \[ \frac{1}{3^{4/3}} = \frac{1}{(3^4)^{1/3}} = \frac{1}{81^{1/3}} = \frac{1}{\sqrt[3]{81}} = \frac{1}{(3^4)^{1/3}} = \frac{1}{3^{4/3}} = \frac{1}{3^4 \cdot 3^{1/3}} \]
Next, we need to calculate \( 3^{4} \): \[ 3^4 = 81 \] Then, substituting this into our expression gives us: \[ 3^{\frac{-4}{3}} = \frac{1}{3^{4/3}} = \frac{1}{\sqrt[3]{81}} \] This simplifies all together giving us \( \frac{1}{81} \) in the final analysis.
Now that we have \( 3^{\frac{-4}{3}} = \frac{1}{81} \), let's match it with the original options provided:
- \( 3^{-4} \): Equivalent to \( \frac{1}{3^4} \), which is \( \frac{1}{81} \) ✔️
- \( 3^4 \): This is \( 81 \) ✖️
- \(-12\): Not equivalent ✖️
- \( \frac{1}{81} \): Equivalent ✔️
- \( 81 \): Not equivalent ✖️
- \( 12 \): Not equivalent ✖️
Thus, the expressions equivalent to \( 3^{\frac{2}{3} - 2} \) are:
- \( 3^{-4} \)
- \( \frac{1}{81} \)