1. **Problem statement:** Find the value of each expression if the result is a real number; otherwise, state NOT REAL. 2. **Recall the rules:** - For any positive number $a$ and rational exponent $\frac{m}{n}$, $a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$. - Negative exponents mean reciprocal: $a^{-k} = \frac{1}{a^k}$. - Even roots of negative numbers are NOT REAL. 3. **Solve (a):** $\left(\frac{16}{25}\right)^{\frac{3}{2}}$ - Rewrite as $\left(\sqrt{\frac{16}{25}}\right)^3$ - $\sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}$ - Now cube: $\left(\frac{4}{5}\right)^3 = \frac{4^3}{5^3} = \frac{64}{125}$ 4. **Solve (b):** $125^{-\frac{2}{3}}$ - Rewrite as $\frac{1}{125^{\frac{2}{3}}}$ - $125^{\frac{2}{3}} = \left(\sqrt[3]{125}\right)^2$ - $\sqrt[3]{125} = 5$ - Square it: $5^2 = 25$ - So, $125^{-\frac{2}{3}} = \frac{1}{25}$ 5. **Solve (c):** $-125^{\frac{2}{3}}$ - Evaluate $125^{\frac{2}{3}}$ first: - $125^{\frac{2}{3}} = \left(\sqrt[3]{125}\right)^2 = 5^2 = 25$ - Apply the negative sign: $-25$ **Final answers:** (a) $\frac{64}{125}$ (b) $\frac{1}{25}$ (c) $-25$