1. **Problem:** Evaluate each power without using a calculator. 2. **Key formula:** For any positive number $a$ and rational exponent $\frac{m}{n}$, we use the rule: $$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$ This means we take the $n$th root of $a$ and then raise it to the $m$th power. 3. **Evaluate each expression:** **a.** $8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$ **b.** $0.027^{\frac{1}{3}} = \sqrt[3]{0.027} = \sqrt[3]{\frac{27}{1000}} = \frac{3}{10} = 0.3$ **c.** $\sqrt[3]{2^6} = 2^{\frac{6}{3}} = 2^2 = 4$ **d.** $\left(\frac{9}{4}\right)^{\frac{3}{2}} = \left(\sqrt{\frac{9}{4}}\right)^3 = \left(\frac{3}{2}\right)^3 = \frac{27}{8}$ **e.** $\sqrt[3]{125^2} = 125^{\frac{2}{3}} = \left(\sqrt[3]{125}\right)^2 = 5^2 = 25$ **f.** $\left(\frac{1}{8}\right)^{\frac{4}{3}} = \left(\sqrt[3]{\frac{1}{8}}\right)^4 = \left(\frac{1}{2}\right)^4 = \frac{1}{16}$ **g.** $0.01^{\frac{3}{2}} = \left(\sqrt{0.01}\right)^3 = (0.1)^3 = 0.001$ **h.** $\left(\frac{8}{27}\right)^{\frac{2}{3}} = \left(\sqrt[3]{\frac{8}{27}}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9}$ **i.** $(-100)^{1.5} = (-100)^{\frac{3}{2}} = \left(\sqrt{-100}\right)^3$ Since $\sqrt{-100}$ is not a real number, this expression is not real-valued. **j.** $\left(\sqrt{\frac{16}{25}}\right)^2 = \left(\frac{4}{5}\right)^2 = \frac{16}{25}$ **k.** $81^{0.75} = 81^{\frac{3}{4}} = \left(\sqrt[4]{81}\right)^3 = 3^3 = 27$ **l.** $\left(\sqrt[3]{\frac{27}{8}}\right)^{\frac{5}{3}} = \left(\frac{3}{2}\right)^{\frac{5}{3}}$ Rewrite as: $$\left(\frac{3}{2}\right)^{\frac{5}{3}} = \left(\left(\frac{3}{2}\right)^{\frac{1}{3}}\right)^5 = \left(\sqrt[3]{\frac{3}{2}}\right)^5$$ This is simplified form; exact numeric value is irrational. 4. **Summary of answers:** - a: 4 - b: 0.3 - c: 4 - d: 27/8 - e: 25 - f: 1/16 - g: 0.001 - h: 4/9 - i: Not real - j: 16/25 - k: 27 - l: $\left(\sqrt[3]{\frac{3}{2}}\right)^5$ These steps show how to evaluate fractional exponents by converting to roots and powers, and checking domain restrictions for real numbers.