1. Problem 44 a) Simplify $$3^{-2} : 3^{-4} - \left(\frac{3}{5}\right)^{-2}$$ 2. Use the rule for division of powers with the same base: $$a^m : a^n = a^{m-n}$$ 3. Calculate $$3^{-2} : 3^{-4} = 3^{-2 - (-4)} = 3^{2} = 9$$ 4. Calculate $$\left(\frac{3}{5}\right)^{-2} = \left(\frac{5}{3}\right)^{2} = \frac{25}{9}$$ 5. Substitute back: $$9 - \frac{25}{9} = \frac{81}{9} - \frac{25}{9} = \frac{56}{9}$$ --- 1. Problem 44 b) Simplify $$\left(\frac{2}{3}\right)^{-2} + 3^{-2} \cdot 3^{-4}$$ 2. Use the rule $$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$$ and $$a^m \cdot a^n = a^{m+n}$$ 3. Calculate $$\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$$ 4. Calculate $$3^{-2} \cdot 3^{-4} = 3^{-2-4} = 3^{-6} = \frac{1}{3^6} = \frac{1}{729}$$ 5. Sum: $$\frac{9}{4} + \frac{1}{729} = \frac{9 \cdot 729}{4 \cdot 729} + \frac{1 \cdot 4}{729 \cdot 4} = \frac{6561}{2916} + \frac{4}{2916} = \frac{6565}{2916}$$ --- 1. Problem 45 a) Simplify $$\left(3^{-2} - 4^{-1}\right) : \left(5 \cdot 36^{-1}\right)$$ 2. Calculate powers: $$3^{-2} = \frac{1}{9}, \quad 4^{-1} = \frac{1}{4}, \quad 36^{-1} = \frac{1}{36}$$ 3. Calculate numerator: $$\frac{1}{9} - \frac{1}{4} = \frac{4}{36} - \frac{9}{36} = -\frac{5}{36}$$ 4. Calculate denominator: $$5 \cdot \frac{1}{36} = \frac{5}{36}$$ 5. Division: $$\frac{-\frac{5}{36}}{\frac{5}{36}} = -\frac{5}{36} \cdot \frac{36}{5} = -\cancel{\frac{5}{36}} \cdot \cancel{\frac{36}{5}} = -1$$ --- 1. Problem 45 b) Simplify $$(-5)^{-5} \cdot 25^{16} \cdot 125^{-4}$$ 2. Express powers with base 5: $$-5 = -1 \cdot 5$$, $$25 = 5^2$$, $$125 = 5^3$$ 3. Calculate powers: $$(-5)^{-5} = (-1)^{-5} \cdot 5^{-5} = -1 \cdot 5^{-5} = -5^{-5}$$ 4. Calculate $$25^{16} = (5^2)^{16} = 5^{32}$$ 5. Calculate $$125^{-4} = (5^3)^{-4} = 5^{-12}$$ 6. Multiply powers of 5: $$-5^{-5} \cdot 5^{32} \cdot 5^{-12} = -5^{-5 + 32 - 12} = -5^{15}$$ --- 1. Problem 46 a) Simplify $$\left(2^{-3} - 2^{-4}\right) : 16^{-1}$$ 2. Calculate powers: $$2^{-3} = \frac{1}{8}, \quad 2^{-4} = \frac{1}{16}, \quad 16^{-1} = \frac{1}{16}$$ 3. Calculate numerator: $$\frac{1}{8} - \frac{1}{16} = \frac{2}{16} - \frac{1}{16} = \frac{1}{16}$$ 4. Division: $$\frac{\frac{1}{16}}{\frac{1}{16}} = 1$$ --- 1. Problem 46 b) Simplify $$4^{-2} \cdot 8^{-5} : 2^{-22}$$ 2. Express powers with base 2: $$4 = 2^2, \quad 8 = 2^3$$ 3. Calculate powers: $$4^{-2} = (2^2)^{-2} = 2^{-4}, \quad 8^{-5} = (2^3)^{-5} = 2^{-15}$$ 4. Multiply numerator: $$2^{-4} \cdot 2^{-15} = 2^{-19}$$ 5. Division: $$\frac{2^{-19}}{2^{-22}} = 2^{-19 - (-22)} = 2^{3} = 8$$ --- 1. Problem 47 a) Simplify $$\left(\frac{4}{25} - \left(\frac{5}{2}\right)^{-3}\right)^{-1}$$ 2. Calculate $$\left(\frac{5}{2}\right)^{-3} = \left(\frac{2}{5}\right)^3 = \frac{8}{125}$$ 3. Calculate inside parentheses: $$\frac{4}{25} - \frac{8}{125} = \frac{20}{125} - \frac{8}{125} = \frac{12}{125}$$ 4. Take inverse: $$\left(\frac{12}{125}\right)^{-1} = \frac{125}{12}$$ --- 1. Problem 47 b) Simplify $$2^{-2} + \left(\frac{1}{2}\right)^{-2} \cdot 2^{-3} + 8^{-2}$$ 2. Calculate powers: $$2^{-2} = \frac{1}{4}, \quad \left(\frac{1}{2}\right)^{-2} = 2^{2} = 4, \quad 2^{-3} = \frac{1}{8}, \quad 8^{-2} = (2^3)^{-2} = 2^{-6} = \frac{1}{64}$$ 3. Calculate middle term: $$4 \cdot \frac{1}{8} = \frac{1}{2}$$ 4. Sum all: $$\frac{1}{4} + \frac{1}{2} + \frac{1}{64} = \frac{16}{64} + \frac{32}{64} + \frac{1}{64} = \frac{49}{64}$$ --- 1. Problem 48 a) Simplify $$5 - \left(\frac{1}{2}\right)^{-1}$$ 2. Calculate $$\left(\frac{1}{2}\right)^{-1} = 2$$ 3. Subtract: $$5 - 2 = 3$$ --- 1. Problem 48 b) Simplify $$3^{-2} - \left(\frac{3}{2}\right)^{-3} \cdot 2^{-4}$$ 2. Calculate powers: $$3^{-2} = \frac{1}{9}, \quad \left(\frac{3}{2}\right)^{-3} = \left(\frac{2}{3}\right)^3 = \frac{8}{27}, \quad 2^{-4} = \frac{1}{16}$$ 3. Multiply: $$\frac{8}{27} \cdot \frac{1}{16} = \frac{8}{432} = \frac{1}{54}$$ 4. Subtract: $$\frac{1}{9} - \frac{1}{54} = \frac{6}{54} - \frac{1}{54} = \frac{5}{54}$$ --- 1. Problem 49 a) Simplify $$\frac{2^{-2} \cdot 5^{3} \cdot 10^{-4}}{2^{-3} \cdot 5^{2} \cdot 10^{-5} \cdot 2^{-2} + 2^{0}}$$ 2. Calculate numerator powers: $$2^{-2} = \frac{1}{4}, \quad 5^{3} = 125, \quad 10^{-4} = \frac{1}{10000}$$ 3. Numerator: $$\frac{1}{4} \cdot 125 \cdot \frac{1}{10000} = \frac{125}{40000} = \frac{125}{40000} = \frac{1}{320}$$ 4. Calculate denominator terms: - $$2^{-3} = \frac{1}{8}$$ - $$5^{2} = 25$$ - $$10^{-5} = \frac{1}{100000}$$ - $$2^{-2} = \frac{1}{4}$$ - $$2^{0} = 1$$ 5. Multiply denominator first term: $$\frac{1}{8} \cdot 25 \cdot \frac{1}{100000} \cdot \frac{1}{4} = \frac{25}{3200000} = \frac{1}{128000}$$ 6. Sum denominator: $$\frac{1}{128000} + 1 = \frac{1 + 128000}{128000} = \frac{128001}{128000}$$ 7. Divide numerator by denominator: $$\frac{\frac{1}{320}}{\frac{128001}{128000}} = \frac{1}{320} \cdot \frac{128000}{128001} = \frac{128000}{320 \cdot 128001} = \frac{400}{128001}$$ --- 1. Problem 49 b) Simplify $$\frac{3^{-3} \cdot 4^{-2} \cdot 10^{-4}}{2^{-5} \cdot 3^{-5} \cdot 10^{-6}}$$ 2. Calculate powers: - $$3^{-3} = \frac{1}{27}$$ - $$4^{-2} = \frac{1}{16}$$ - $$10^{-4} = \frac{1}{10000}$$ - $$2^{-5} = \frac{1}{32}$$ - $$3^{-5} = \frac{1}{243}$$ - $$10^{-6} = \frac{1}{1000000}$$ 3. Numerator: $$\frac{1}{27} \cdot \frac{1}{16} \cdot \frac{1}{10000} = \frac{1}{4320000}$$ 4. Denominator: $$\frac{1}{32} \cdot \frac{1}{243} \cdot \frac{1}{1000000} = \frac{1}{7776000000}$$ 5. Divide numerator by denominator: $$\frac{\frac{1}{4320000}}{\frac{1}{7776000000}} = \frac{1}{4320000} \cdot 7776000000 = 1800$$