Реши примеры на упрощение выражений и сравнение чисел из задач 74, 75, 76, 77, 78, 79, 80, 81, 82

74. 1) $a^{\sqrt{2}} \cdot a^{1-\sqrt{2}} = a^{\sqrt{2} + 1 - \sqrt{2}} = a^1 = a$ 2) $a^{\sqrt{3}-1} \cdot a^{\sqrt{3}+1} = a^{\sqrt{3}-1 + \sqrt{3}+1} = a^{2\sqrt{3}}$ 3) $(b^{\sqrt{3}})^{\sqrt{3}} : b^2 = b^{\sqrt{3} \cdot \sqrt{3}} : b^2 = b^3 : b^2 = b^{3-2} = b^1 = b$ 75. Чтобы сравнить числа, нужно привести их к одному показателю корня или основанию степени. 1) $\sqrt[2]{2}$ и $\sqrt[3]{8}$. $\sqrt[2]{2} = 2^{\frac{1}{2}}$, $\sqrt[3]{8} = 8^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} = 2^{3 \cdot \frac{1}{3}} = 2^1 = 2$. Значит, $\sqrt[2]{2} < \sqrt[3]{8}$. 2) $\sqrt{5}$ и $\sqrt{7}$. Т.к. $5 < 7$, то $\sqrt{5} < \sqrt{7}$. 76. 1) $(\frac{1}{16})^{-0.75} + 810000^{0.25} - (\frac{19}{32})^{\frac{1}{5}} = (\frac{1}{2^4})^{-\frac{3}{4}} + (3^4 \cdot 10^4)^{0.25} - (\frac{19}{32})^{\frac{1}{5}} = (2^{-4})^{-\frac{3}{4}} + (30^4)^{\frac{1}{4}} - (\frac{19}{32})^{\frac{1}{5}} = 2^3 + 30 - (\frac{19}{32})^{\frac{1}{5}} = 8 + 30 - (\frac{19}{32})^{\frac{1}{5}} = 38 - (\frac{19}{32})^{\frac{1}{5}}$ 2) $27^{\frac{2}{3}} - (-2)^{-2} + (3\frac{3}{8})^{-\frac{1}{2}} = (3^3)^{\frac{2}{3}} - (\frac{1}{-2})^{2} + (\frac{27}{8})^{-\frac{1}{2}} = 3^2 - \frac{1}{4} + (\frac{3^3}{2^3})^{-\frac{1}{2}} = 9 - \frac{1}{4} + (\frac{3}{2})^{-3 \cdot -\frac{1}{2}} = 9 - \frac{1}{4} + (\frac{3}{2})^{\frac{3}{2}} = 9 - \frac{1}{4} + (\frac{2}{3})^{\frac{3}{2}}$ 3) $(0,001)^{-\frac{1}{3}} - 2^{-2} \cdot 64^{\frac{2}{3}} - 8^{-\frac{1}{3}} = (\frac{1}{1000})^{-\frac{1}{3}} - \frac{1}{2^2} \cdot (2^6)^{\frac{2}{3}} - (2^3)^{-\frac{1}{3}} = (\frac{1}{10^3})^{-\frac{1}{3}} - \frac{1}{4} \cdot 2^4 - 2^{-1} = 10 - \frac{1}{4} \cdot 16 - \frac{1}{2} = 10 - 4 - \frac{1}{2} = 6 - \frac{1}{2} = 5\frac{1}{2}$ 4) $(-0,5)^{-4} - 625^{0.25} - (2\frac{1}{4})^{-\frac{1}{2}} = (-\frac{1}{2})^{-4} - (5^4)^{\frac{1}{4}} - (\frac{9}{4})^{-\frac{1}{2}} = (-2)^4 - 5 - (\frac{3^2}{2^2})^{-\frac{1}{2}} = 16 - 5 - (\frac{3}{2})^{-2 \cdot -\frac{1}{2}} = 11 - (\frac{3}{2}) = 11 - \frac{3}{2} = 11 - 1\frac{1}{2} = 9\frac{1}{2}$ 77. 1) $(a^4)^{-\frac{3}{4}} \cdot (b^{-\frac{2}{3}})^{-6} = a^{4 \cdot -\frac{3}{4}} \cdot b^{-\frac{2}{3} \cdot -6} = a^{-3} \cdot b^4 = \frac{b^4}{a^3}$ 2) $((\frac{a^6}{b^{-3}})^4)^{\frac{1}{12}} = (\frac{a^{6 \cdot 4}}{b^{-3 \cdot 4}})^{\frac{1}{12}} = (\frac{a^{24}}{b^{-12}})^{\frac{1}{12}} = (a^{24} \cdot b^{12})^{\frac{1}{12}} = a^{24 \cdot \frac{1}{12}} \cdot b^{12 \cdot \frac{1}{12}} = a^2 \cdot b$ 78. 1) $\frac{a^{\frac{4}{5}}}{a^{\frac{3}{5}} (a^{-\frac{1}{5}} + a^{\frac{2}{5}})} = \frac{a^{\frac{4}{5}}}{a^{\frac{3}{5} - \frac{1}{5}} + a^{\frac{3}{5} + \frac{2}{5}}} = \frac{a^{\frac{4}{5}}}{a^{\frac{2}{5}} + a^{\frac{5}{5}}} = \frac{a^{\frac{4}{5}}}{a^{\frac{2}{5}} + a} $ 2) $\frac{b^{\frac{1}{5}}}{b^{\frac{2}{3}} (\sqrt[5]{b^4} - \sqrt[5]{b^{-1}})} = \frac{b^{\frac{1}{5}}}{b^{\frac{2}{3}} (b^{\frac{4}{5}} - b^{-\frac{1}{5}})} = \frac{b^{\frac{1}{5}}}{b^{\frac{2}{3} + \frac{4}{5}} - b^{\frac{2}{3} - \frac{1}{5}}} = \frac{b^{\frac{1}{5}}}{b^{\frac{10+12}{15}} - b^{\frac{10-3}{15}}} = \frac{b^{\frac{1}{5}}}{b^{\frac{22}{15}} - b^{\frac{7}{15}}}$ 3) $\frac{a^{\frac{5}{3}} b^{-1} - a^{-\frac{1}{3}}}{\sqrt[3]{a^2}} = \frac{a^{\frac{5}{3}} b^{-1} - a^{-\frac{1}{3}}}{a^{\frac{2}{3}}} = a^{\frac{5}{3} - \frac{2}{3}} b^{-1} - a^{-\frac{1}{3} - \frac{2}{3}} = a^{\frac{3}{3}} b^{-1} - a^{-\frac{3}{3}} = a \cdot \frac{1}{b} - a^{-1} = \frac{a}{b} - \frac{1}{a}$ 4) $\frac{a^{\frac{1}{3}} \sqrt{b} + b^{\frac{1}{3}} \sqrt{a}}{\sqrt[6]{a} + \sqrt[6]{b}} = \frac{a^{\frac{1}{3}} b^{\frac{1}{2}} + b^{\frac{1}{3}} a^{\frac{1}{2}}}{a^{\frac{1}{6}} + b^{\frac{1}{6}}} = \frac{a^{\frac{2}{6}} b^{\frac{3}{6}} + b^{\frac{2}{6}} a^{\frac{3}{6}}}{a^{\frac{1}{6}} + b^{\frac{1}{6}}} = \frac{a^{\frac{2}{6}} b^{\frac{2}{6}} (b^{\frac{1}{6}} + a^{\frac{1}{6}})}{a^{\frac{1}{6}} + b^{\frac{1}{6}}} = a^{\frac{2}{6}} b^{\frac{2}{6}} = a^{\frac{1}{3}} b^{\frac{1}{3}} = (ab)^{\frac{1}{3}} = \sqrt[3]{ab}$ 79. 1) $(2^{\frac{2}{3}} \cdot 3^{-\frac{1}{3}} - 3^{\frac{5}{3}} \cdot 2^{-\frac{1}{3}})^{\frac{3}{6}} = (2^{\frac{2}{3}} \cdot 3^{-\frac{1}{3}} - 3^{\frac{5}{3}} \cdot 2^{-\frac{1}{3}})^{\frac{1}{2}} = (\frac{2^{\frac{2}{3}}}{3^{\frac{1}{3}}} - \frac{3^{\frac{5}{3}}}{2^{-\frac{1}{3}}})^{\frac{1}{2}} = (\frac{2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} - 3^{\frac{5}{3}} \cdot 3^{\frac{1}{3}}}{3^{\frac{1}{3}} \cdot 2^{\frac{1}{3}}})^{\frac{1}{2}} = (\frac{2 - 3^2}{3^{\frac{1}{3}} \cdot 2^{\frac{1}{3}}})^{\frac{1}{2}} = (\frac{2-9}{3^{\frac{1}{3}} \cdot 2^{\frac{1}{3}}})^{\frac{1}{2}} = (\frac{-7}{(3 \cdot 2)^{\frac{1}{3}}})^{\frac{1}{2}} = (\frac{-7}{6^{\frac{1}{3}}})^{\frac{1}{2}}$ 2) $(54^{\frac{1}{4}} : 24^{-\frac{1}{4}} - 24^{\frac{3}{4}} : 54^{\frac{3}{4}}) \cdot \sqrt[4]{1000} = (\frac{54^{\frac{1}{4}}}{24^{-\frac{1}{4}}} - \frac{24^{\frac{3}{4}}}{54^{\frac{3}{4}}}) \cdot \sqrt[4]{1000} = (54^{\frac{1}{4}} \cdot 24^{\frac{1}{4}} - (\frac{24}{54})^{\frac{3}{4}}) \cdot \sqrt[4]{1000} = ((54 \cdot 24)^{\frac{1}{4}} - (\frac{24}{54})^{\frac{3}{4}}) \cdot \sqrt[4]{1000} = ((2 \cdot 3^3 \cdot 2^3 \cdot 3)^{\frac{1}{4}} - (\frac{2^3 \cdot 3}{2 \cdot 3^3})^{\frac{3}{4}}) \cdot \sqrt[4]{1000} = ((2^4 \cdot 3^4)^{\frac{1}{4}} - (\frac{2^2}{3^2})^{\frac{3}{4}}) \cdot \sqrt[4]{1000} = (2 \cdot 3 - (\frac{2}{3})^{\frac{6}{4}}) \cdot \sqrt[4]{1000} = (6 - (\frac{2}{3})^{\frac{3}{2}}) \cdot \sqrt[4]{1000}$ 80. 1) $a^{\frac{1}{6}} \sqrt[3]{a \sqrt{a}} = a^{\frac{1}{6}} \sqrt[3]{a \cdot a^{\frac{1}{2}}} = a^{\frac{1}{6}} \sqrt[3]{a^{\frac{3}{2}}} = a^{\frac{1}{6}} a^{\frac{3}{2} \cdot \frac{1}{3}} = a^{\frac{1}{6}} a^{\frac{1}{2}} = a^{\frac{1}{6} + \frac{1}{2}} = a^{\frac{1+3}{6}} = a^{\frac{4}{6}} = a^{\frac{2}{3}}$ 2) $b^{\frac{1}{12}} \sqrt[3]{b \sqrt[4]{b \sqrt{b}}} = b^{\frac{1}{12}} \sqrt[3]{b \sqrt[4]{b \cdot b^{\frac{1}{2}}}} = b^{\frac{1}{12}} \sqrt[3]{b \sqrt[4]{b^{\frac{3}{2}}}} = b^{\frac{1}{12}} \sqrt[3]{b \cdot b^{\frac{3}{2} \cdot \frac{1}{4}}} = b^{\frac{1}{12}} \sqrt[3]{b \cdot b^{\frac{3}{8}}} = b^{\frac{1}{12}} \sqrt[3]{b^{\frac{11}{8}}} = b^{\frac{1}{12}} b^{\frac{11}{8} \cdot \frac{1}{3}} = b^{\frac{1}{12}} b^{\frac{11}{24}} = b^{\frac{2}{24} + \frac{11}{24}} = b^{\frac{13}{24}}$ 3) $(\sqrt[3]{ab^{-2}} + (ab)^{-\frac{1}{6}}) \sqrt[6]{ab^4} = (\sqrt[3]{\frac{a}{b^2}} + \frac{1}{\sqrt[6]{ab}}) \sqrt[6]{ab^4} = (\frac{\sqrt[6]{a^2}}{\sqrt[6]{b^4}} + \frac{1}{\sqrt[6]{ab}}) \sqrt[6]{ab^4} = (\frac{\sqrt[6]{a^2}}{\sqrt[6]{b^4}} \cdot \sqrt[6]{ab^4} + \frac{1}{\sqrt[6]{ab}} \cdot \sqrt[6]{ab^4}) = (\sqrt[6]{\frac{a^3 b^4}{b^4}} + \sqrt[6]{\frac{ab^4}{ab}}) = \sqrt[6]{a^3} + \sqrt[6]{b^3} = \sqrt{a} + \sqrt{b}$ 4) $(\sqrt{a} + \sqrt[3]{b}) (a^{\frac{2}{3}} + b - \sqrt[3]{a} \sqrt{b}) = a - a^{\frac{4}{6}} b^{\frac{3}{6}} + a^{\frac{2}{3}} b^{\frac{1}{2}} + a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{3}} b^{\frac{3}{6}} + b^{\frac{4}{3}} = a - a^{\frac{2}{3}} b^{\frac{1}{2}} + a^{\frac{2}{3}} b^{\frac{1}{2}} + a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{3}} b^{\frac{1}{2}} + b^{\frac{4}{3}} = a + b^{\frac{4}{3}} + a^{\frac{2}{3}} b^{\frac{1}{3}} - a^{\frac{1}{3}} b^{\frac{1}{2}}$ 81. 1) $(1 - 2 \sqrt{\frac{b}{a}} + \frac{b}{a}) : (\frac{a^2 - b^2}{a}) = (1 - 2 \sqrt{\frac{b}{a}} + (\sqrt{\frac{b}{a}})^2) : (\frac{(a-b)(a+b)}{a}) = (1 - \sqrt{\frac{b}{a}})^2 \cdot \frac{a}{(a-b)(a+b)} = (\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a}})^2 \cdot \frac{a}{(a-b)(a+b)} = \frac{(\sqrt{a} - \sqrt{b})^2}{a} \cdot \frac{a}{(a-b)(a+b)} = \frac{(\sqrt{a} - \sqrt{b})^2}{(a-b)(a+b)} = \frac{(\sqrt{a} - \sqrt{b})^2}{(\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b})(a+b)} = \frac{\sqrt{a} - \sqrt{b}}{(\sqrt{a} + \sqrt{b})(a+b)}$ 2) $(a^{\frac{3}{2}} + b^{\frac{3}{2}}) : (2 \sqrt{\frac{a}{b}} + 3 + \sqrt{\frac{b}{a}}) = (a^{\frac{3}{2}} + b^{\frac{3}{2}}) : (2 \frac{\sqrt{a}}{\sqrt{b}} + 3 + \frac{\sqrt{b}}{\sqrt{a}}) = (a^{\frac{3}{2}} + b^{\frac{3}{2}}) : (\frac{2a + 3 \sqrt{ab} + b}{\sqrt{ab}}) = (a^{\frac{3}{2}} + b^{\frac{3}{2}}) \cdot (\frac{\sqrt{ab}}{2a + 3 \sqrt{ab} + b})$ 3) $\frac{a^{\frac{4}{5}} - a^{-\frac{1}{5}}}{a^{\frac{4}{5}} - a^{-\frac{1}{5}}} - \frac{b^2 - b^{\frac{1}{2}}}{b^2 + b^{\frac{1}{2}}} = 1 - \frac{b^{\frac{1}{2}} (b^{\frac{3}{2}} - 1)}{b^{\frac{1}{2}} (b^{\frac{3}{2}} + 1)} = 1 - \frac{b^{\frac{3}{2}} - 1}{b^{\frac{3}{2}} + 1} = \frac{b^{\frac{3}{2}} + 1 - b^{\frac{3}{2}} + 1}{b^{\frac{3}{2}} + 1} = \frac{2}{b^{\frac{3}{2}} + 1}$ 4) $\frac{\sqrt{a} - a^{-\frac{1}{2}} b}{1 - \sqrt{a^{-1} b}} : \frac{\sqrt[6]{a} + a^{-\frac{1}{3}} \sqrt{b}}{1} = \frac{\sqrt{a} - \frac{b}{\sqrt{a}}}{1 - \sqrt{\frac{b}{a}}} : (\sqrt[6]{a} + \frac{b^{\frac{1}{2}}}{a^{\frac{1}{3}}}) = \frac{\frac{a - b}{\sqrt{a}}}{\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a}}} : (\sqrt[6]{a} + \frac{b^{\frac{1}{2}}}{a^{\frac{1}{3}}}) = \frac{a - b}{\sqrt{a} - \sqrt{b}} : (a^{\frac{1}{6}} + \frac{b^{\frac{1}{2}}}{a^{\frac{1}{3}}}) = \frac{(\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b})}{\sqrt{a} - \sqrt{b}} : (a^{\frac{1}{6}} + \frac{b^{\frac{1}{2}}}{a^{\frac{1}{3}}}) = (\sqrt{a} + \sqrt{b}) : (a^{\frac{1}{6}} + \frac{b^{\frac{1}{2}}}{a^{\frac{1}{3}}})$ 82. 1) $\frac{m^{\sqrt{5}} \cdot n^{\sqrt{3}}}{(mn)^{2 + \sqrt{3}}} = \frac{m^{\sqrt{5}} \cdot n^{\sqrt{3}}}{m^{2 + \sqrt{3}} n^{2 + \sqrt{3}}} = \frac{m^{\sqrt{5}} \cdot n^{\sqrt{3}}}{m^{2 + \sqrt{3}} n^{2 + \sqrt{3}}} = m^{\sqrt{5} - 2 - \sqrt{3}} n^{\sqrt{3} - 2 - \sqrt{3}} = m^{\sqrt{5} - 2 - \sqrt{3}} n^{-2}$ 2) $\frac{x^{\sqrt{7}} \cdot y^{\sqrt{7} + 1}}{(xy)^{\sqrt{7}}} = \frac{x^{\sqrt{7}} \cdot y^{\sqrt{7} + 1}}{x^{\sqrt{7}} y^{\sqrt{7}}} = \frac{x^{\sqrt{7}} \cdot y^{\sqrt{7}} \cdot y}{x^{\sqrt{7}} y^{\sqrt{7}}} = y$ 3) $(a^{\sqrt{2}} - b^{\sqrt{3}})(a^{\sqrt{2}} + b^{\sqrt{3}}) = (a^{\sqrt{2}})^2 - (b^{\sqrt{3}})^2 = a^{2\sqrt{2}} - b^{2\sqrt{3}}$ 4) $(2a^{-0.5} - \frac{1}{3}b^{-\sqrt{3}})(\frac{1}{3}b^{-\sqrt{3}} + 2a^{-0.5}) = (2a^{-0.5} - \frac{1}{3}b^{-\sqrt{3}})(2a^{-0.5} + \frac{1}{3}b^{-\sqrt{3}}) = (2a^{-0.5})^2 - (\frac{1}{3}b^{-\sqrt{3}})^2 = 4a^{-1} - \frac{1}{9}b^{-2\sqrt{3}} = \frac{4}{a} - \frac{1}{9b^{2\sqrt{3}}}$