Use the properties of exponents to determine whether the pair of expressions are equivalent. \[ \left(\frac{1}{3}\right)^{4} \text { and }\left(\frac{1}{9}\right)^{2} \] The expressions \( \square \) equal. The expression \( \left(\frac{1}{3}\right)^{4} \) can be rewritten as 3 \( \square \) The expression \( \left(\frac{1}{9}\right)^{2} \) can be rewritten as 9 \( \square \) , which is equal to 3 \( \square \) The expressions have the same base, \( \square \) equal. (Type integers or simplified fractions.)

To determine whether the expressions \(\left(\frac{1}{3}\right)^{4}\) and \(\left(\frac{1}{9}\right)^{2}\) are equivalent, we will simplify both expressions step by step. 1. **Simplifying \(\left(\frac{1}{3}\right)^{4}\)**: \[ \left(\frac{1}{3}\right)^{4} = \frac{1^{4}}{3^{4}} = \frac{1}{81} \] 2. **Simplifying \(\left(\frac{1}{9}\right)^{2}\)**: \[ \left(\frac{1}{9}\right)^{2} = \frac{1^{2}}{9^{2}} = \frac{1}{81} \] Now we can compare the two simplified expressions: - \(\left(\frac{1}{3}\right)^{4} = \frac{1}{81}\) - \(\left(\frac{1}{9}\right)^{2} = \frac{1}{81}\) Since both expressions are equal to \(\frac{1}{81}\), we conclude that: - The expressions \(\left(\frac{1}{3}\right)^{4}\) and \(\left(\frac{1}{9}\right)^{2}\) are equivalent. Now, filling in the blanks: - The expressions \( \text{are} \) equal. - The expression \( \left(\frac{1}{3}\right)^{4} \) can be rewritten as \( \frac{1}{81} \). - The expression \( \left(\frac{1}{9}\right)^{2} \) can be rewritten as \( \frac{1}{81} \), which is equal to \( \frac{1}{81} \). - The expressions have the same base, \( \text{are} \) equal.