1. **Problem:** Evaluate each power without using a calculator. 2. **Formula and rules:** For any positive number $a$ and rational exponent $m/n$, we use $$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m.$$ Negative bases with fractional exponents require careful consideration of the root's parity. 3. **Step-by-step evaluation:** **a.** $16^{\frac{2}{3}} = \left(16^{\frac{1}{3}}\right)^2 = (\sqrt[3]{16})^2$. Since $16 = 2^4$, $\sqrt[3]{16} = 2^{\frac{4}{3}}$, so $16^{\frac{2}{3}} = (2^{\frac{4}{3}})^2 = 2^{\frac{8}{3}} = 2^{2 + \frac{2}{3}} = 4 \times 2^{\frac{2}{3}}$ (exact form) or approximate as $\approx 6.35$. **b.** $81^{\frac{1}{4} \times 3} = 81^{\frac{3}{4}} = \left(81^{\frac{1}{4}}\right)^3$. Since $81 = 3^4$, $81^{\frac{1}{4}} = 3$, so $81^{\frac{3}{4}} = 3^3 = 27$. **c.** $32^{\frac{4}{5}} = \left(32^{\frac{1}{5}}\right)^4$. Since $32 = 2^5$, $32^{\frac{1}{5}} = 2$, so $32^{\frac{4}{5}} = 2^4 = 16$. **d.** $(9^{\frac{4}{2}})^5 = (9^2)^5 = 81^5$. Since $9^2 = 81$, the expression is $81^5$ (exact form). **e.** $0.008^{\frac{3}{2}} = \left(0.008^{\frac{1}{2}}\right)^3 = (\sqrt{0.008})^3$. Note $0.008 = \frac{8}{1000} = \frac{2^3}{10^3} = (\frac{2}{10})^3 = 0.2^3$, so $\sqrt{0.008} = 0.2^{\frac{3}{2}} = 0.2^{1.5}$. Alternatively, $\sqrt{0.008} = \sqrt{\frac{8}{1000}} = \frac{\sqrt{8}}{\sqrt{1000}} = \frac{2\sqrt{2}}{10\sqrt{10}}$. Then cube it for exact form or approximate $\approx 0.000715$. **f.** $(625^{\frac{1}{4}})^3$. Since $625 = 5^4$, $625^{\frac{1}{4}} = 5$, so the expression is $5^3 = 125$. **g.** $\frac{36^{\frac{3}{2}}}{81}$. Since $36^{\frac{3}{2}} = (36^{\frac{1}{2}})^3 = 6^3 = 216$, and $81 = 9^2$, the expression is $\frac{216}{81} = \frac{216}{81} = \frac{72}{27} = \frac{8}{3} \approx 2.6667$. **h.** $(-27)^{\frac{3}{4}}$. Since the denominator 4 is even, the fourth root of a negative number is not real. So this expression is not a real number. **Final answers:** **a.** $16^{\frac{2}{3}} = 4 \times 2^{\frac{2}{3}}$ (exact), approx $6.35$ **b.** $27$ **c.** $16$ **d.** $81^5$ **e.** approx $0.000715$ **f.** $125$ **g.** $\frac{8}{3}$ **h.** Not a real number (no real fourth root of negative number)