Помоги упростить выражения и вычислить значения выражений в заданиях 74-82

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 1) Сравним $\sqrt[3]{2}$ и $\sqrt{3}$. Возведём оба числа в степень 6 (наименьшее общее кратное показателей корней): $(\sqrt[3]{2})^6 = 2^{6/3} = 2^2 = 4$ $(\sqrt{3})^6 = 3^{6/2} = 3^3 = 27$ Так как $4 < 27$, то $\sqrt[3]{2} < \sqrt{3}$. 2) Сравним $\sqrt[5]{4}$ и $\sqrt[7]{4}$. Возведём оба числа в степень 35 (наименьшее общее кратное показателей корней): $(\sqrt[5]{4})^{35} = 4^{35/5} = 4^7 = 16384$ $(\sqrt[7]{4})^{35} = 4^{35/7} = 4^5 = 1024$ Так как $16384 > 1024$, то $\sqrt[5]{4} > \sqrt[7]{4}$. 76 1) $(\frac{1}{16})^{-0.75} + 810000^{0.25} - (7\frac{19}{32})^{\frac{1}{5}} = (2^{-4})^{-3/4} + (3^4 \cdot 10^4)^{1/4} - (\frac{243}{32})^{1/5} = 2^3 + 3 \cdot 10 - (\frac{3^5}{2^5})^{1/5} = 8 + 30 - \frac{3}{2} = 38 - 1.5 = 36.5$ 2) $27^{\frac{2}{3}} - (-2)^{-2} + (3\frac{3}{8})^{-\frac{1}{2}} = (3^3)^{\frac{2}{3}} - (\frac{1}{-2})^2 + (\frac{27}{8})^{-\frac{1}{2}} = 3^2 - \frac{1}{4} + (\frac{3^3}{2^3})^{-\frac{1}{2}} = 9 - 0.25 + (\frac{3}{2})^{-3/2} = 9 - 0.25 + (\frac{2}{3})^{\frac{3}{2}} = 9 - 0.25 + \frac{2\sqrt{2}}{3\sqrt{3}} = 8.75 + \frac{2\sqrt{2}}{3\sqrt{3}}$ 3) $(0.001)^{-\frac{1}{3}} - 2^{-2} \cdot 64^{\frac{2}{3}} \cdot 8 = (\frac{1}{1000})^{-\frac{1}{3}} - (\frac{1}{2})^2 \cdot (2^6)^{\frac{2}{3}} \cdot 2^3 = 1000^{\frac{1}{3}} - \frac{1}{4} \cdot 2^4 \cdot 2^3 = 10 - \frac{1}{4} \cdot 16 \cdot 8 = 10 - 4 \cdot 8 = 10 - 32 = -22$ 4) $(-0.5)^{-4} - 625^{0.25} - (2\frac{1}{4})^{-\frac{1}{2}} = (-\frac{1}{2})^{-4} - (5^4)^{\frac{1}{4}} - (\frac{9}{4})^{-\frac{1}{2}} = (-2)^4 - 5 - (\frac{3}{2})^{-1} = 16 - 5 - \frac{2}{3} = 11 - \frac{2}{3} = 10\frac{1}{3}$ 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) $(\frac{a^6}{b^{-3}})^{\frac{4}{3}})^{\frac{1}{12}} = (a^6 \cdot b^3)^{\frac{4}{3} \cdot \frac{1}{12}} = (a^6 \cdot b^3)^{\frac{1}{9}} = a^{\frac{6}{9}} \cdot b^{\frac{3}{9}} = a^{\frac{2}{3}} \cdot b^{\frac{1}{3}} = \sqrt[3]{a^2b}$ 78 1) $\frac{a^{\frac{4}{3}}(a^{-\frac{1}{3}} + a^{\frac{2}{3}})}{a^{\frac{1}{4}}(a^{\frac{3}{4}} + a^{-\frac{1}{4}})} = \frac{a^{\frac{4}{3}-\frac{1}{3}} + a^{\frac{4}{3}+\frac{2}{3}}}{a^{\frac{1}{4}+\frac{3}{4}} + a^{\frac{1}{4}-\frac{1}{4}}} = \frac{a^1 + a^2}{a^1 + a^0} = \frac{a+a^2}{a+1} = \frac{a(1+a)}{a+1} = a$ 2) $\frac{b^5(\sqrt{b^4} - \sqrt{b^{-1}})}{b^3(\sqrt{b} - \sqrt{b^{-2}})} = \frac{b^5(b^2 - b^{-\frac{1}{2}})}{b^3(b^{\frac{1}{2}} - b^{-1})} = \frac{b^5(b^2 - \frac{1}{\sqrt{b}})}{b^3(\sqrt{b} - \frac{1}{b})} = b^2 \cdot \frac{b^2 - \frac{1}{\sqrt{b}}}{\sqrt{b} - \frac{1}{b}} = b^2 \cdot \frac{\frac{b^{2.5}-1}{\sqrt{b}}}{\frac{b^{1.5}-1}{b}} = b^2 \cdot \frac{b(b^{2.5}-1)}{\sqrt{b}(b^{1.5}-1)} = b^{1.5} \cdot \frac{(b^{2.5}-1)}{(b^{1.5}-1)}$ 3) $\frac{a^{\frac{5}{3}}b^{-1} - a^{-\frac{1}{3}}}{\sqrt[3]{a^2}} = \frac{a^{-\frac{1}{3}}(a^{\frac{6}{3}}b^{-1} - 1)}{a^{\frac{2}{3}}} = \frac{a^{-\frac{1}{3}}(a^2b^{-1} - 1)}{a^{\frac{2}{3}}} = a^{-\frac{1}{3} - \frac{2}{3}}(a^2b^{-1} - 1) = a^{-1}(a^2b^{-1} - 1) = \frac{a^2}{ab} - \frac{1}{a} = \frac{a}{b} - \frac{1}{a}$ 4) $\frac{a^{\frac{1}{3}}\sqrt{b} + b^{\frac{1}{3}}\sqrt{a}}{\sqrt[6]{a} + \sqrt[6]{b}} = \frac{a^{\frac{1}{3}}b^{\frac{1}{2}} + b^{\frac{1}{3}}a^{\frac{1}{2}}}{a^{\frac{1}{6}} + b^{\frac{1}{6}}} = \frac{a^{\frac{1}{3}}b^{\frac{1}{2}} + b^{\frac{1}{3}}a^{\frac{1}{2}}}{a^{\frac{1}{6}} + b^{\frac{1}{6}}} = \frac{a^{\frac{1}{3}}b^{\frac{1}{2}} + a^{\frac{1}{2}}b^{\frac{1}{3}}}{a^{\frac{1}{6}} + b^{\frac{1}{6}}} = \frac{a^{\frac{1}{3}}b^{\frac{1}{3}}(b^{\frac{1}{6}} + a^{\frac{1}{6}})}{a^{\frac{1}{6}} + b^{\frac{1}{6}}} = a^{\frac{1}{3}}b^{\frac{1}{3}} = \sqrt[3]{ab}$ 79 1) $(2^{\frac{5}{3}} \cdot 3^{-\frac{1}{3}} - 3^{\frac{5}{3}} \cdot 2^{-\frac{1}{3}})^{\frac{1}{6}} = (\frac{2^{\frac{5}{3}}}{3^{\frac{1}{3}}} - \frac{3^{\frac{5}{3}}}{2^{\frac{1}{3}}})^{\frac{1}{6}} = (\frac{2^{\frac{5}{3}} \cdot 2^{\frac{1}{3}} - 3^{\frac{5}{3}} \cdot 3^{\frac{1}{3}}}{3^{\frac{1}{3}} \cdot 2^{\frac{1}{3}}})^{\frac{1}{6}} = (\frac{2^2 - 3^2}{3^{\frac{1}{3}} \cdot 2^{\frac{1}{3}}})^{\frac{1}{6}} = (\frac{4-9}{3^{\frac{1}{3}} \cdot 2^{\frac{1}{3}}})^{\frac{1}{6}} = (\frac{-5}{3^{\frac{1}{3}} \cdot 2^{\frac{1}{3}}})^{\frac{1}{6}}$ 2) $(54 : 24 - 24 : 54)^{\frac{1}{4}} \cdot \sqrt[4]{1000} = (\frac{54}{24} - \frac{24}{54})^{\frac{1}{4}} \cdot \sqrt[4]{1000} = (\frac{9}{4} - \frac{4}{9})^{\frac{1}{4}} \cdot \sqrt[4]{1000} = (\frac{81-16}{36})^{\frac{1}{4}} \cdot \sqrt[4]{1000} = (\frac{65}{36})^{\frac{1}{4}} \cdot \sqrt[4]{1000}$ 80 1) $a^{\frac{9}{6}} \sqrt{a\sqrt{a}} = a^{\frac{3}{2}} \cdot (a \cdot a^{\frac{1}{2}})^{\frac{1}{2}} = a^{\frac{3}{2}} \cdot (a^{\frac{3}{2}})^{\frac{1}{2}} = a^{\frac{3}{2}} \cdot a^{\frac{3}{4}} = a^{\frac{6}{4} + \frac{3}{4}} = a^{\frac{9}{4}}$ 2) $b^{\frac{1}{12}} \sqrt[3]{b \sqrt[4]{b}} = b^{\frac{1}{12}} \cdot (b \cdot b^{\frac{1}{4}})^{\frac{1}{3}} = b^{\frac{1}{12}} \cdot (b^{\frac{5}{4}})^{\frac{1}{3}} = b^{\frac{1}{12}} \cdot b^{\frac{5}{12}} = b^{\frac{1}{12} + \frac{5}{12}} = b^{\frac{6}{12}} = b^{\frac{1}{2}} = \sqrt{b}$ 3) $(\sqrt[3]{ab^{-2}} + (ab)^{-\frac{1}{6}})^{\frac{1}{6}} \sqrt{ab^4} = (a^{\frac{1}{3}}b^{-\frac{2}{3}} + a^{-\frac{1}{6}}b^{-\frac{1}{6}})^{\frac{1}{6}} \cdot a^{\frac{1}{2}}b^2 = (a^{\frac{2}{6}}b^{-\frac{4}{6}} + a^{-\frac{1}{6}}b^{-\frac{1}{6}})^{\frac{1}{6}} \cdot a^{\frac{1}{2}}b^2 = (a^{-\frac{1}{6}}b^{-\frac{1}{6}}(a^{\frac{3}{6}}b^{-\frac{3}{6}} + 1))^{\frac{1}{6}} \cdot a^{\frac{1}{2}}b^2 = (a^{-\frac{1}{6}}b^{-\frac{1}{6}}(a^{\frac{1}{2}}b^{-\frac{1}{2}} + 1))^{\frac{1}{6}} \cdot a^{\frac{1}{2}}b^2$ 4) $(\sqrt{a} + \sqrt[3]{b})(a^{\frac{2}{3}} + b^{\frac{2}{3}} - \sqrt[3]{ab}) = (a^{\frac{1}{2}} + b^{\frac{1}{3}})(a^{\frac{2}{3}} + b^{\frac{2}{3}} - a^{\frac{1}{3}}b^{\frac{1}{3}}) = a^{\frac{1}{2}}a^{\frac{2}{3}} + a^{\frac{1}{2}}b^{\frac{2}{3}} - a^{\frac{1}{2}}a^{\frac{1}{3}}b^{\frac{1}{3}} + b^{\frac{1}{3}}a^{\frac{2}{3}} + b^{\frac{1}{3}}b^{\frac{2}{3}} - b^{\frac{1}{3}}a^{\frac{1}{3}}b^{\frac{1}{3}} = a^{\frac{7}{6}} + a^{\frac{1}{2}}b^{\frac{2}{3}} - a^{\frac{5}{6}}b^{\frac{1}{3}} + a^{\frac{2}{3}}b^{\frac{1}{3}} + b - a^{\frac{1}{3}}b^{\frac{2}{3}}$ 81 1) $(1 - 2\sqrt{\frac{b}{a}} + \frac{b}{a}) : (\frac{a^{\frac{1}{2}} - b^{\frac{1}{2}}}{a^{\frac{1}{2}} + b^{\frac{1}{2}}})^2 = (1 - 2\sqrt{\frac{b}{a}} + \frac{b}{a}) : (\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}})^2 = (1 - \sqrt{\frac{b}{a}})^2 : (\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}})^2$ 2) $(a^{\frac{3}{2}} + b^{\frac{3}{2}}) : (2 + 3\sqrt{\frac{a}{b}} + 3\sqrt{\frac{b}{a}}) = (a^{\frac{3}{2}} + b^{\frac{3}{2}}) : (2 + 3\sqrt{\frac{a}{b}} + 3\sqrt{\frac{b}{a}})$ 3) $\frac{a^{-\frac{1}{4}} - a^{\frac{5}{4}}}{a^{\frac{1}{4}} - a^{-\frac{1}{4}}} - \frac{b^{-\frac{1}{2}} - b^{\frac{3}{2}}}{b^{\frac{1}{2}} + b^{-\frac{1}{2}}} = \frac{a^{-\frac{1}{4}}(1 - a^{\frac{6}{4}})}{a^{-\frac{1}{4}}(a^{\frac{2}{4}} - 1)} - \frac{b^{-\frac{1}{2}}(1 - b^2)}{b^{-\frac{1}{2}}(b + 1)} = \frac{1 - a^{\frac{3}{2}}}{\sqrt{a} - 1} - \frac{1 - b^2}{b+1} = \frac{1 - a^{\frac{3}{2}}}{\sqrt{a} - 1} - \frac{(1 - b)(1+b)}{b+1} = \frac{1 - a^{\frac{3}{2}}}{\sqrt{a} - 1} - (1-b)$ 4) $\frac{\sqrt{a} - a^{-\frac{1}{2}}b}{1-\sqrt{a^{-1}}b} \cdot \frac{a^{\frac{3}{2}} - a^{\frac{1}{2}}b}{\sqrt[6]{a} + a^{-\frac{1}{3}}\sqrt{b}} = \frac{\sqrt{a} - \frac{b}{\sqrt{a}}}{1-\sqrt{\frac{b}{a}}} \cdot \frac{a^{\frac{3}{2}} - a^{\frac{1}{2}}b}{\sqrt[6]{a} + \frac{\sqrt{b}}{a^{\frac{1}{3}}}} = \frac{\frac{a-b}{\sqrt{a}}}{\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}}} \cdot \frac{a^{\frac{1}{2}}(a-b)}{\sqrt[6]{a} + \frac{\sqrt{b}}{a^{\frac{1}{3}}}} = \frac{(a-b)\sqrt{a}}{\sqrt{a} - \sqrt{b}} \cdot \frac{\sqrt{a}(a-b)}{\sqrt[6]{a} + \frac{\sqrt{b}}{a^{\frac{1}{3}}}} = \frac{a-b}{\sqrt{a} - \sqrt{b}} \cdot \frac{\sqrt{a}(a-b)}{\sqrt[6]{a} + \frac{\sqrt{b}}{a^{\frac{1}{3}}}}$ 82 Допущение: в задании 1 пропущена степень у $n$ во втором множителе числителя. Будем считать, что там $\sqrt[3]{n}$. 1) $\frac{m^{\sqrt{3}} \cdot n^{\sqrt{3}}}{(mn)^{2+\sqrt{3}}} = \frac{(mn)^{\sqrt{3}}}{(mn)^{2+\sqrt{3}}} = (mn)^{\sqrt{3}-(2+\sqrt{3})} = (mn)^{-2} = \frac{1}{(mn)^2}$ 2) $\frac{x^{\sqrt{7}} \cdot y^{\sqrt{7}+1}}{(xy)^{\sqrt{7}}} = \frac{x^{\sqrt{7}} \cdot y^{\sqrt{7}} \cdot y}{(xy)^{\sqrt{7}}} = \frac{(xy)^{\sqrt{7}} \cdot y}{(xy)^{\sqrt{7}}} = y$ 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) $(2a^{-0.5} - \frac{1}{3}b^{-\sqrt{3}})(\frac{1}{3}b^{-\sqrt{3}} + 2a^{-0.5}) = (2a^{-0.5})^2 - (\frac{1}{3}b^{-\sqrt{3}})^2 = 4a^{-1} - \frac{1}{9}b^{-2\sqrt{3}} = \frac{4}{a} - \frac{1}{9b^{2\sqrt{3}}}$