1. **State the problem:** Complete the table by converting radical expressions to rational exponents and evaluating them to two decimal places. 2. **Recall the formula:** A radical expression $\sqrt[n]{a^m}$ can be written as $a^{\frac{m}{n}}$. 3. **Evaluate each expression:** - For $\sqrt[9]{22^{10}}$, written as $22^{\frac{10}{9}}$, calculate $22^{1.111\ldots}$. - For $\sqrt[8]{10^{7}}$, written as $10^{\frac{7}{8}}$, calculate $10^{0.875}$. - For $\sqrt[6]{5^{7}}$, written as $5^{\frac{7}{6}}$, calculate $5^{1.166\ldots}$. 4. **Calculations:** $$22^{\frac{10}{9}} \approx 25.02$$ $$10^{\frac{7}{8}} \approx 7.50$$ $$5^{\frac{7}{6}} \approx 7.23$$ 5. **Final table:** | Written in radical notation | Written using rational exponents | Evaluated to two decimal places | |----------------------------|----------------------------------|---------------------------------| | $\sqrt{2}$ | $2^{\frac{1}{2}}$ | 1.41 | | $\sqrt[9]{22^{10}}$ | $22^{\frac{10}{9}}$ | 25.02 | | $\sqrt[8]{10^{7}}$ | $10^{\frac{7}{8}}$ | 7.50 | | $\sqrt[6]{5^{7}}$ | $5^{\frac{7}{6}}$ | 7.23 |