1. Problem: Evaluate each expression without using a calculator. 2. Recall the rules of exponents: - $a^{m} \times a^{n} = a^{m+n}$ - $\frac{a^{m}}{a^{n}} = a^{m-n}$ - $(a^{m})^{n} = a^{m \times n}$ - $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ 3. Solve each part step-by-step: a. $\frac{16^{7/2}}{2^{6}} = \frac{(2^{4})^{7/2}}{2^{6}} = \frac{2^{4 \times \frac{7}{2}}}{2^{6}} = \frac{2^{14}}{2^{6}} = 2^{14-6} = 2^{8} = 256$ b. $27^{2/3} = (3^{3})^{2/3} = 3^{3 \times \frac{2}{3}} = 3^{2} = 9$ c. $\frac{5^{2/3}}{\sqrt{5}} = \frac{5^{2/3}}{5^{1/2}} = 5^{2/3 - 1/2} = 5^{\frac{4}{6} - \frac{3}{6}} = 5^{1/6}$ d. $\frac{16^{3/4}}{16^{1/2}} = 16^{3/4 - 1/2} = 16^{\frac{3}{4} - \frac{2}{4}} = 16^{1/4} = (2^{4})^{1/4} = 2^{1} = 2$ e. $\frac{8^{3/5}}{\sqrt[3]{4}} = \frac{(2^{3})^{3/5}}{4^{1/3}} = \frac{2^{9/5}}{(2^{2})^{1/3}} = \frac{2^{9/5}}{2^{2/3}} = 2^{9/5 - 2/3} = 2^{\frac{27}{15} - \frac{10}{15}} = 2^{17/15}$ f. $\frac{7^{2/3}}{7^{1/3}} = 7^{2/3 - 1/3} = 7^{1/3}$ g. $5^{2/3} \times 5^{4/5} = 5^{2/3 + 4/5} = 5^{\frac{10}{15} + \frac{12}{15}} = 5^{22/15}$ h. $9^{1/4} \times 9^{1/4} = 9^{1/4 + 1/4} = 9^{1/2} = (3^{2})^{1/2} = 3^{1} = 3$ i. $8 \times 2^{-2} = 2^{3} \times 2^{-2} = 2^{3 - 2} = 2^{1} = 2$ j. $6^{4/3} \times 6^{2/3} = 6^{4/3 + 2/3} = 6^{6/3} = 6^{2} = 36$ k. $2^{-2} \times 16 = 2^{-2} \times 2^{4} = 2^{-2 + 4} = 2^{2} = 4$ l. $8^{5/3} \times 8^{-1/3} = 8^{5/3 - 1/3} = 8^{4/3} = (2^{3})^{4/3} = 2^{4} = 16$ m. $\frac{3^{1/3} \times 3^{2/3}}{3} = \frac{3^{1/3 + 2/3}}{3^{1}} = \frac{3^{1}}{3^{1}} = 1$ n. $\frac{7^{5/7} \times 7^{6/7}}{7^{2}} = \frac{7^{(5/7 + 6/7)}}{7^{2}} = \frac{7^{11/7}}{7^{2}} = 7^{11/7 - 2} = 7^{11/7 - 14/7} = 7^{-3/7}$ o. $\frac{2^{3/8}}{\sqrt{8}} = \frac{2^{3/8}}{8^{1/2}} = \frac{2^{3/8}}{(2^{3})^{1/2}} = \frac{2^{3/8}}{2^{3/2}} = 2^{3/8 - 3/2} = 2^{3/8 - 12/8} = 2^{-9/8}$ p. $(5^{1/3})^{5} \times 5^{-1/2} = 5^{5/3} \times 5^{-1/2} = 5^{5/3 - 1/2} = 5^{\frac{10}{6} - \frac{3}{6}} = 5^{7/6}$ q. $\frac{8^{3} + 7}{27^{1/3}} = \frac{512 + 7}{3} = \frac{519}{3} = 173$ r. $\frac{9^{2/3} \times 3^{2/3}}{3^{1/3} \times 3^{1/6}} = \frac{(3^{2})^{2/3} \times 3^{2/3}}{3^{1/3 + 1/6}} = \frac{3^{4/3} \times 3^{2/3}}{3^{1/2}} = \frac{3^{(4/3 + 2/3)}}{3^{1/2}} = \frac{3^{2}}{3^{1/2}} = 3^{2 - 1/2} = 3^{3/2}$ Final answers: a. 256 b. 9 c. $5^{1/6}$ d. 2 e. $2^{17/15}$ f. $7^{1/3}$ g. $5^{22/15}$ h. 3 i. 2 j. 36 k. 4 l. 16 m. 1 n. $7^{-3/7}$ o. $2^{-9/8}$ p. $5^{7/6}$ q. 173 r. $3^{3/2}$