1. **Problem Statement:** Simplify the given expressions using index laws. 2. **Index Laws Used:** - Power of a power: $\left(a^m\right)^n = a^{mn}$ - Product of powers with same base: $a^m \times a^n = a^{m+n}$ - Quotient of powers with same base: $\frac{a^m}{a^n} = a^{m-n}$ - Any nonzero number to the zero power: $a^0 = 1$ --- ### Example 20b a. Simplify $\left(2^3\right)^2 \times \left(2^5\right)^3$ Step 1: Apply power of a power: $2^{3 \times 2} \times 2^{5 \times 3} = 2^6 \times 2^{15}$ Step 2: Multiply powers with same base: $2^{6+15} = 2^{21}$ b. Simplify $\left(5^2\right)^6 \times \left(5^3\right)^2$ Step 1: $5^{2 \times 6} \times 5^{3 \times 2} = 5^{12} \times 5^6$ Step 2: $5^{12+6} = 5^{18}$ c. Simplify $\left(7^4\right)^2 \times \left(7^5\right)^3$ Step 1: $7^{4 \times 2} \times 7^{5 \times 3} = 7^8 \times 7^{15}$ Step 2: $7^{8+15} = 7^{23}$ d. Simplify $\left(3^3\right)^6 \times \left(3^2\right)^2$ Step 1: $3^{3 \times 6} \times 3^{2 \times 2} = 3^{18} \times 3^4$ Step 2: $3^{18+4} = 3^{22}$ e. Simplify $\left(2^3\right)^2 \times \left(2^4\right)^3$ Step 1: $2^{3 \times 2} \times 2^{4 \times 3} = 2^6 \times 2^{12}$ Step 2: $2^{6+12} = 2^{18}$ f. Simplify $\left(11^2\right)^6 \times \left(11^2\right)^3$ Step 1: $11^{2 \times 6} \times 11^{2 \times 3} = 11^{12} \times 11^6$ Step 2: $11^{12+6} = 11^{18}$ g. Simplify $\frac{\left(5^3\right)^4}{5^2}$ Step 1: $\frac{5^{3 \times 4}}{5^2} = \frac{5^{12}}{5^2}$ Step 2: $5^{12-2} = 5^{10}$ h. Simplify $\frac{\left(11^7\right)^2}{\left(11^3\right)^2}$ Step 1: $\frac{11^{7 \times 2}}{11^{3 \times 2}} = \frac{11^{14}}{11^6}$ Step 2: $11^{14-6} = 11^8$ i. Simplify $\frac{\left(11^5\right)^3}{11^2}$ Step 1: $\frac{11^{5 \times 3}}{11^2} = \frac{11^{15}}{11^2}$ Step 2: $11^{15-2} = 11^{13}$ j. Simplify $\frac{\left(2^2\right)^{10}}{\left(2^3\right)^2}$ Step 1: $\frac{2^{2 \times 10}}{2^{3 \times 2}} = \frac{2^{20}}{2^6}$ Step 2: $2^{20-6} = 2^{14}$ k. Simplify $\frac{3^{20}}{\left(3^3\right)^5}$ Step 1: $\frac{3^{20}}{3^{3 \times 5}} = \frac{3^{20}}{3^{15}}$ Step 2: $3^{20-15} = 3^5$ l. Simplify $\frac{\left(11^2\right)^{10}}{\left(11^3\right)^6}$ Step 1: $\frac{11^{2 \times 10}}{11^{3 \times 6}} = \frac{11^{20}}{11^{18}}$ Step 2: $11^{20-18} = 11^2$ --- ### Example 19 a. Simplify $3^0$ Any nonzero number to zero power is 1: $3^0 = 1$ b. Simplify $5^0 = 1$ c. Simplify $3^0 + 5^0 = 1 + 1 = 2$ d. Simplify $3 + 5^0 = 3 + 1 = 4$ e. Simplify $(3 + 5)^0 = 8^0 = 1$ f. Simplify $3^0 \times 5^0 = 1 \times 1 = 1$ g. Simplify $(3 \times 5)^0 = 15^0 = 1$ h. Simplify $3 \times 5^0 = 3 \times 1 = 3$ i. Simplify $(35 \times 53)^0 = (1855)^0 = 1$ --- **Final answers:** Example 20b: a) $2^{21}$ b) $5^{18}$ c) $7^{23}$ d) $3^{22}$ e) $2^{18}$ f) $11^{18}$ g) $5^{10}$ h) $11^8$ i) $11^{13}$ j) $2^{14}$ k) $3^5$ l) $11^2$ Example 19: a) 1 b) 1 c) 2 d) 4 e) 1 f) 1 g) 1 h) 3 i) 1