1. **Rewrite each expression in negative index form:** (a) $\frac{1}{6^2} = 6^{-2}$ (b) $\frac{1}{7^5} = 7^{-5}$ (c) $\frac{1}{20^{17}} = 20^{-17}$ (d) $\frac{1}{8} = 8^{-1}$ (e) $\frac{1}{a^2} = a^{-2}$ (f) $\frac{1}{x} = x^{-1}$ (g) $\frac{1}{(2y)^3} = (2y)^{-3}$ 2. **Write the following in fractional form (using positive exponents):** (a) $7^{-2} = \frac{1}{7^2}$ (b) $2^{-5} = \frac{1}{2^5}$ (c) $3^{-2} = \frac{1}{3^2}$ (d) $2.5^{-3} = \frac{1}{2.5^3}$ (e) $2^{-3} \times 2 = \frac{1}{2^3} \times 2 = \frac{2}{2^3} = \frac{2}{8} = \frac{1}{4}$ (f) $3^{-5} \div 3^{-3} = 3^{-5 - (-3)} = 3^{-2} = \frac{1}{3^2}$ (g) $x^{-5} \div x^{-2} = x^{-5 - (-2)} = x^{-3} = \frac{1}{x^3}$ (h) $5b^{-3} \times 3b = 15 b^{-3+1} = 15 b^{-2} = \frac{15}{b^2}$ (i) $(2x)^{-1} \div x^{-5} = (2x)^{-1} \times x^5 = \frac{x^5}{2x} = \frac{x^{5-1}}{2} = \frac{x^4}{2}$ (j) $5 a^{-1} \times a^{-3} = 5 a^{-1-3} = 5 a^{-4} = \frac{5}{a^4}$ (k) $(2x)^0 \div x^3 = 1 \div x^3 = \frac{1}{x^3}$ (l) $4 y^{-1} \times 2 y^{-3} = 8 y^{-1-3} = 8 y^{-4} = \frac{8}{y^4}$ (m) $(2a)^{-1} \times a^{-3} = \frac{1}{2a} \times a^{-3} = \frac{1}{2} a^{-1-3} = \frac{1}{2 a^{4}}$ (n) $10 x^4 \times 2 x^{-1} = 20 x^{4-1} = 20 x^{3}$ (o) $10^5 \div 10^5 \times 2.5^{-2} = 10^{5-5} \times \frac{1}{2.5^2} = 1 \times \frac{1}{6.25} = \frac{1}{6.25}$ (p) $5^{-3} \times 5^{-2} \times 5^4 \times (5h)^0 = 5^{-3-2+4} \times 1 = 5^{-1} = \frac{1}{5}$ **Final answers:** (a) $6^{-2}$ (b) $7^{-5}$ (c) $20^{-17}$ (d) $8^{-1}$ (e) $a^{-2}$ (f) $x^{-1}$ (g) $(2y)^{-3}$ (a) $\frac{1}{7^2}$ (b) $\frac{1}{2^5}$ (c) $\frac{1}{3^2}$ (d) $\frac{1}{2.5^3}$ (e) $\frac{1}{4}$ (f) $\frac{1}{3^2}$ (g) $\frac{1}{x^3}$ (h) $\frac{15}{b^2}$ (i) $\frac{x^4}{2}$ (j) $\frac{5}{a^4}$ (k) $\frac{1}{x^3}$ (l) $\frac{8}{y^4}$ (m) $\frac{1}{2 a^4}$ (n) $20 x^{3}$ (o) $\frac{1}{6.25}$ (p) $\frac{1}{5}$