Преобразуй выражение 2² * 2⁻¹

1. $2^2 \cdot 2^{-1} = 2^{2-1} = 2^1 = 2$ 2. $8 \cdot 2^{-4} = 2^3 \cdot 2^{-4} = 2^{3-4} = 2^{-1} = \frac{1}{2}$ 3. $2^5:(2^2 \cdot 2^5) = 2^5 : 2^{2+5} = 2^5 : 2^7 = 2^{5-7} = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}$ 4. $\frac{(-2)^5}{3^3} = \frac{-32}{27} = -1 \frac{5}{27}$ 5. $\frac{5^4 \cdot 2^{10}}{10} = \frac{5^4 \cdot 2^{10}}{5 \cdot 2} = 5^{4-1} \cdot 2^{10-1} = 5^3 \cdot 2^9 = 125 \cdot 512 = 64000$ 6. $\left( \frac{x^4}{x^3 \cdot x^2} \right)^{-2} : \frac{x^3 \cdot x^2}{x} = \left( \frac{x^4}{x^5} \right)^{-2} : \frac{x^5}{x} = \left( \frac{1}{x} \right)^{-2} : x^4 = x^2 : x^4 = \frac{x^2}{x^4} = \frac{1}{x^2}$ 7. $(0,2x^{-3}y^{-2})^2 \cdot \left( \frac{x^2}{2y^3} \right)^{-2} = (\frac{1}{5}x^{-3}y^{-2})^2 \cdot \left( \frac{2y^3}{x^2} \right)^{2} = \frac{1}{25}x^{-6}y^{-4} \cdot \frac{4y^6}{x^4} = \frac{4}{25}x^{-10}y^{2} = \frac{4y^2}{25x^{10}}$ 8. $(4^{-1})^2 \cdot 2^5 \cdot \left( \frac{1}{16} \right)^{-3} \cdot (8^{-2})^5 \cdot (64^3) = (2^{-2})^2 \cdot 2^5 \cdot (2^{-4})^{-3} \cdot (2^3)^{-10} \cdot (2^6)^3 = 2^{-4} \cdot 2^5 \cdot 2^{12} \cdot 2^{-30} \cdot 2^{18} = 2^{-4+5+12-30+18} = 2^1 = 1$ 9. $2^{-2} \cdot \frac{1}{4} \cdot \left( \frac{1}{2} \right)^{-3} \cdot 4 : \frac{1}{25} = 2^{-2} \cdot 2^{-2} \cdot 2^{3} \cdot 2^2 \cdot 25 = 2^{-2-2+3+2} \cdot 25 = 2^{-1} \cdot 25 = \frac{25}{2} = 12,5$ 10. $\left( -2 \frac{1}{2} \right)^{-3} : (0,25)^2 \cdot ((-5)^{-2})^2 = \left( -\frac{5}{2} \right)^{-3} : (\frac{1}{4})^2 \cdot (\frac{1}{25})^2 = \left( -\frac{2}{5} \right)^{3} : \frac{1}{16} \cdot \frac{1}{625} = -\frac{8}{125} \cdot 16 \cdot 625 = -\frac{8 \cdot 16 \cdot 625}{125} = -8 \cdot 16 \cdot 5 = -640$ 11. $\left( -\frac{7x^2}{3y^4} \right)^{-3} \cdot \left( \frac{9y^2}{49x^4} \right)^{-2} = \left( -\frac{3y^4}{7x^2} \right)^{3} \cdot \left( \frac{49x^4}{9y^2} \right)^{2} = -\frac{27y^{12}}{343x^6} \cdot \frac{2401x^8}{81y^4} = -\frac{27 \cdot 2401}{343 \cdot 81} \cdot \frac{y^{12}x^8}{x^6y^4} = -\frac{7 \cdot 3}{1 \cdot 9}x^{8-6}y^{12-4} = -\frac{7}{3}x^2y^8$ 12. $\left( \frac{x^5}{y^2} \right)^{2} : \left( \frac{x^3}{3y^7} \right)^{-2} = \frac{x^{10}}{y^4} : \left( \frac{3y^7}{x^3} \right)^{2} = \frac{x^{10}}{y^4} : \frac{9y^{14}}{x^6} = \frac{x^{10}}{y^4} \cdot \frac{x^6}{9y^{14}} = \frac{x^{16}}{9y^{18}}$ 13. $\left( \frac{1}{2}x^{-1}y^3 \right)^{-3} : (x^2 \cdot y^{-4}) = \left( 2x^{1}y^{-3} \right)^{3} : (x^2 \cdot y^{-4}) = 8x^3y^{-9} : x^2y^{-4} = 8x^{3-2}y^{-9-(-4)} = 8xy^{-5} = \frac{8x}{y^5}$ 14. $\left( \frac{1}{6}x^{-7}y^3 \right)^{-2} \cdot \left( \frac{x^3}{y^2} \right)^{-2} \cdot \left( \frac{2x^4}{y^3} \right)^{4} = (6x^{7}y^{-3})^{2} \cdot \left( \frac{y^2}{x^3} \right)^{2} \cdot \frac{16x^{16}}{y^{12}} = 36x^{14}y^{-6} \cdot \frac{y^4}{x^6} \cdot \frac{16x^{16}}{y^{12}} = \frac{36 \cdot 16 x^{14+16}y^4}{x^6y^{6+12}} = \frac{576 x^{30}y^4}{x^6y^{18}} = 576 x^{30-6} y^{4-18} = 576 x^{24} y^{-14} = \frac{576x^{24}}{y^{14}}$ 17. $\sqrt{\frac{54}{24}} = \sqrt{\frac{9}{4}} = \frac{3}{2} = 1,5$ 18. $\left( \frac{1}{5} \right)^{-3} : \sqrt{25} = 5^3 : 5 = 5^2 = 25$ 19. $\sqrt{49 \cdot 36 \cdot 100} = \sqrt{49} \cdot \sqrt{36} \cdot \sqrt{100} = 7 \cdot 6 \cdot 10 = 420$ 20. $\frac{21}{\sqrt{3}} \cdot \sqrt{\frac{1}{3}} = \frac{21}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}} = \frac{21}{3} = 7$ 21. $\frac{\sqrt[4]{16 \cdot 81} \cdot \sqrt{12}}{\sqrt{3}} = \frac{\sqrt[4]{2^4 \cdot 3^4} \cdot \sqrt{4 \cdot 3}}{\sqrt{3}} = \frac{2 \cdot 3 \cdot 2 \cdot \sqrt{3}}{\sqrt{3}} = 12$ 22. $\sqrt[3]{27 \cdot 81^3} \cdot \left( \frac{1}{2} \right)^{\frac{2}{3}} \cdot \sqrt{4} = \sqrt[3]{3^3 \cdot (3^4)^3} \cdot \left( \frac{1}{2} \right)^{\frac{2}{3}} \cdot 2 = \sqrt[3]{3^3 \cdot 3^{12}} \cdot \left( \frac{1}{2} \right)^{\frac{2}{3}} \cdot 2 = \sqrt[3]{3^{15}} \cdot \left( \frac{1}{2} \right)^{\frac{2}{3}} \cdot 2 = 3^5 \cdot \left( \frac{1}{2} \right)^{\frac{2}{3}} \cdot 2 = 243 \cdot \left( \frac{1}{2} \right)^{\frac{2}{3}} \cdot 2 $ 23. $\frac{\sqrt[3]{9^2} \cdot \left( \frac{1}{3} \right)^6}{\sqrt{3} \cdot 27^{\frac{1}{3}}} = \frac{\sqrt[3]{(3^2)^2} \cdot \left( \frac{1}{3} \right)^6}{\sqrt{3} \cdot (3^3)^{\frac{1}{3}}} = \frac{\sqrt[3]{3^4} \cdot \left( \frac{1}{3} \right)^6}{\sqrt{3} \cdot 3} = \frac{3^{\frac{4}{3}} \cdot 3^{-6}}{3^{\frac{1}{2}} \cdot 3} = \frac{3^{\frac{4}{3}-6}}{3^{\frac{1}{2}+1}} = \frac{3^{\frac{4-18}{3}}}{3^{\frac{1+2}{2}}} = \frac{3^{-\frac{14}{3}}}{3^{\frac{3}{2}}} = 3^{-\frac{14}{3}-\frac{3}{2}} = 3^{-\frac{28+9}{6}} = 3^{-\frac{37}{6}} = \frac{1}{3^{\frac{37}{6}}}$ 24. $(0,75 \cdot \sqrt{9}) : \left( 0,25 \cdot \sqrt{2\frac{2}{3}} \right) = (\frac{3}{4} \cdot 3) : (\frac{1}{4} \cdot \sqrt{\frac{8}{3}}) = \frac{9}{4} : (\frac{1}{4} \cdot \sqrt{\frac{4 \cdot 2}{3}}) = \frac{9}{4} : (\frac{1}{4} \cdot \frac{2\sqrt{2}}{\sqrt{3}}) = \frac{9}{4} \cdot \frac{4\sqrt{3}}{2\sqrt{2}} = \frac{9\sqrt{3}}{2\sqrt{2}} = \frac{9\sqrt{6}}{4}$ 25. $\sqrt{\sqrt{3}} \cdot (\sqrt{\sqrt{3}} : \sqrt{\sqrt{\sqrt{3}}})^2 = \sqrt[4]{3} \cdot (\sqrt[4]{3} : \sqrt[8]{3})^2 = \sqrt[4]{3} \cdot (3^{\frac{1}{4}} : 3^{\frac{1}{8}})^2 = 3^{\frac{1}{4}} \cdot (3^{\frac{1}{4}-\frac{1}{8}})^2 = 3^{\frac{1}{4}} \cdot (3^{\frac{2-1}{8}})^2 = 3^{\frac{1}{4}} \cdot (3^{\frac{1}{8}})^2 = 3^{\frac{1}{4}} \cdot 3^{\frac{2}{8}} = 3^{\frac{1}{4}} \cdot 3^{\frac{1}{4}} = 3^{\frac{1}{4} + \frac{1}{4}} = 3^{\frac{2}{4}} = 3^{\frac{1}{2}} = \sqrt{3}$ 26. $\sqrt{a^4} : \sqrt{(-a)^4} = a^2 : (-a)^2 = a^2 : a^2 = 1$ 27. $\sqrt{x} : \sqrt{\frac{x}{y}} = \sqrt{x} \cdot \sqrt{\frac{y}{x}} = \sqrt{x \cdot \frac{y}{x}} = \sqrt{y}$