Simplify the expression. \[ \left(\frac{4}{\frac{4}{5}} \cdot a^{-\frac{1}{3}}\right)^{2} \] Write your answer without using negative exponents. Assume that all variables are positive real numbers. \( \square \)

To simplify the expression \(\left(\frac{4}{\frac{4}{5}} \cdot a^{-\frac{1}{3}}\right)^{2}\), we start by simplifying the fraction \(\frac{4}{\frac{4}{5}}\). Calculating it step by step: \[ \frac{4}{\frac{4}{5}} = 4 \cdot \frac{5}{4} = 5 \] Now, substituting this back into the expression, we have: \[ \left(5 \cdot a^{-\frac{1}{3}}\right)^{2} \] Next, we apply the exponent \(2\) to both parts of the product: \[ (5)^{2} \cdot \left(a^{-\frac{1}{3}}\right)^{2} = 25 \cdot a^{-\frac{2}{3}} \] Now, we need to express \(a^{-\frac{2}{3}}\) in terms of positive exponents: \[ a^{-\frac{2}{3}} = \frac{1}{a^{\frac{2}{3}}} \] Thus, putting this together: \[ 25 \cdot a^{-\frac{2}{3}} = \frac{25}{a^{\frac{2}{3}}} \] Finally, we can express our answer without negative exponents: \[ \boxed{\frac{25}{a^{\frac{2}{3}}}} \]