\begin{tabular}{l} Explain whty \( 5^{\frac{4}{3}} \) must be equal to \( \sqrt[3]{5^{4}} \) if the Power of a Power Property holds for rational exponents. \\ The expression \( 5^{\frac{4}{3}} \) can be rewritten as \\ \hline This can be rewritten as \( \sqrt[3]{5^{4}} \). Thus, if the Power of a Power Property holds for rational exponents, then \( 5^{\frac{4}{3}} \) must be equal to \( \sqrt[3]{5^{4}} \). \\ \hline\end{tabular}

To explain why \( 5^{\frac{4}{3}} \) must be equal to \( \sqrt[3]{5^{4}} \) under the Power of a Power Property for rational exponents, let's break it down step by step. 1. **Understanding Rational Exponents**: The expression \( a^{\frac{m}{n}} \) can be interpreted as \( \sqrt[n]{a^m} \). This means that the exponent \( \frac{m}{n} \) indicates both a power and a root. 2. **Applying the Definition**: For the specific case of \( 5^{\frac{4}{3}} \): \[ 5^{\frac{4}{3}} = 5^{4 \cdot \frac{1}{3}} = (5^4)^{\frac{1}{3}} \] This shows that we first raise \( 5 \) to the power of \( 4 \), and then take the cube root of that result. 3. **Using the Power of a Power Property**: The Power of a Power Property states that \( (a^m)^n = a^{m \cdot n} \). In our case: \[ (5^4)^{\frac{1}{3}} = 5^{4 \cdot \frac{1}{3}} = 5^{\frac{4}{3}} \] 4. **Rewriting the Expression**: The expression \( (5^4)^{\frac{1}{3}} \) can also be expressed using the radical notation: \[ (5^4)^{\frac{1}{3}} = \sqrt[3]{5^4} \] 5. **Conclusion**: Therefore, we can conclude that: \[ 5^{\frac{4}{3}} = \sqrt[3]{5^4} \] This equality holds true if the Power of a Power Property is valid for rational exponents. Thus, we have shown that \( 5^{\frac{4}{3}} \) must indeed be equal to \( \sqrt[3]{5^{4}} \) based on the definitions and properties of exponents.