Вычисления и преобразования со степенями и корнями. Необходимо преобразовать выражения....

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Свойство степеней: $\boxed{a^b\cdot a^c=a^{b+c}}\\ \boxed{a^b:a^c=a^{b-c}}\\ \boxed{a^{-1}= \frac{1}{a} }\\ \boxed{(a^b)^c=a^{b\cdot c}}$

$\displaystyle 1)\,\, 2^2\cdot 2^3=2^{2+3}=2^5=32.\\ \\ 2)\,\,8\cdot 2^{-4}=2^3\cdot 2^{-4}=2^{3-4}=2^{-1}= \frac{1}{2} .\\ \\ 3)\,\, 2^{-5}:(2^3:2^6)=2^{-5}:2^{3-6}=2^{-5}:2^{-3}=2^{-5+3}=2^{-2}= \frac{1}{2^2}= \frac{1}{4} \\ \\ 4) \frac{(-2)^8\cdot 5^3}{5^4\cdot2^{10}\cdot10} = \frac{(-1)^8\cdot 2^8\cdot 5^3}{5^4\cdot2^{10}\cdot 10} = \frac{1}{5\cdot 2^2\cdot 10} = \frac{1}{100} \\ \\ 5)\,\, (b^{-2}\cdot b)^2:b^{-3}=(b^{-2+1})^2:b^{-3}=(b^{-1})^2:b^{-3}=b^{-2}:b^{-3}=b^{-2+3}=b$

$6)\,\, \displaystyle \bigg( \frac{x^4}{x^3\cdot x^2} \bigg)^{-2}\cdot \frac{x^3:x^2}{x} =\bigg( \frac{1}{x} \bigg)^{-2}\cdot \frac{x}{x} =x^2$

$7)\,\, \displaystyle (0.2x^{-3}y^{-2})^2\cdot \bigg( \frac{x^{-2}}{2y^3} \bigg)^{-2}=0.2^2\cdot x^{-6}y^{-4}\cdot x^4\cdot 2^2\cdot y^{6}=\\ \\ =0.04\cdot x^{-6+4}\cdot 4\cdot y^{6-4}=0.16x^{-2}y^2= \frac{0.16y^2}{x^2}$

$\displaystyle 8)\,\, (4^{-1})^2\cdot 2^5\cdot\bigg( \frac{1}{16}\bigg)^3\cdot(8^{-2})^5\cdot (64^2)^3=4^{-2}\cdot 2^5\cdot 16^{-3}\cdot 8^{-10}\cdot 64^6=\\ \\ =(2^2)^{-2}\cdot 2^5\cdot (2^4)^{-3}\cdot (2^3)^{-10}\cdot (2^6)^6=2^{-4}\cdot 2^5\cdot 2^{-12}\cdot 2^{-30}\cdot 2^{36}=\\ \\ =2^{-4+5-12-30+36}=2^{-5}= \frac{1}{2^5}= \frac{1}{32}$

$9)\,\,\displaystyle 2^2\cdot \frac{1}{4} \cdot\bigg( \frac{1}{2}\bigg)^{-3}\cdot 4: \frac{1}{2^5} =2^2\cdot 2^{-2}\cdot 2^3\cdot 2^2\cdot 2^5=2^{10}=1024$

$10)\,\, \displaystyle \bigg(-2 \frac{1}{2} \bigg)^3:(0.25)^2\cdot ((-5)^2)^2=\bigg( -\frac{5}{2}\bigg)^3 :\bigg( \frac{1}{4} \bigg)^2\cdot 5^4=\\ \\ =- \frac{5^3}{2^3} \cdot2^4\cdot 5^4=-10\cdot 5^6=156250$

$\displaystyle 11)\,\, \bigg(- \frac{7x^2}{3y^4}\bigg)^{-3}:\bigg( \frac{9y^2}{49x^4} \bigg)^{-2}=- \frac{7^{-3}x^{-6}}{3^{-3}y^{-12}} : \frac{9^{-2}y^{-4}}{49^{-2}x^{-8}}=\\ \\ = -\frac{7^{-3}\cdot 7^4\cdot x^{-6}\cdot x^8}{3^{-3}\cdot y^{-12}\cdot 3^4\cdot y^4 } =- \frac{7x^2y^8}{3}$

$12)\,\, \displaystyle \bigg( \frac{x^5}{y^2} \bigg)^{-2}:\bigg( \frac{x^3}{3y^7} \bigg)^{-2}= \frac{x^{-10}}{y^{-4}} : \frac{x^{-6}}{3^{-2}y^{-14}} = \frac{x^{-10}\cdot x^6}{y^{-4}\cdot 3^2\cdot y^{14}} = \frac{1}{9x^4y^{10}}$

$13)\,\,\displaystyle \bigg( \frac{1}{2} x^{-1}y^3\bigg)^{-3}:(x^{-2}\cdot y^{-8})=2^3\cdot x^3y^{-9}\cdot x^2\cdot y^8= \frac{8x^5}{y}$

$14)\,\, \displaystyle \bigg( \frac{1}{6} x^{-7}y^3\bigg)^{-2}\cdot \bigg( \frac{x^2}{y^2} \bigg)^{-2}\cdot \bigg( \frac{2x^2}{y^3} \bigg)^{-4}=36x^{14}y^{-6}x^{-4}y^4 \frac{1}{16} x^{-8}y^{12}=\\ \\ = \frac{9x^2y^{10}}{4}$

$15)\,\,\displaystyle \frac{2\cdot3^{20}-5\cdot3^{19}}{(-9)^9} = -\frac{3^{19}(2\cdot 3-5)}{3^{18}}=-3$

<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle+16%29%5C%2C%5C%2C++%5Cfrac%7B%283%5Ccdot+2%5E%7B20%7D%2B7%5Ccdot+2%5E%7B19%7D%29%5Ccdot+52%7D%7B%28-1%29%5E7%5Ccdot%2813%5Ccdot8%5E4%29%5E2%7D+%3D+-%5Cfrac%7B2%5E%7B21%7D%283%5Ccdot+2%2B7%29%5Ccdot+13%7D%7B13%5E2%5Ccdot2%5E%7B24%7D%7D+%3D+-%5Cfrac%7B13%5E2%7D%7B13%5E2%5Ccdot+2%5E3%7D+%3D-+%5Cfrac%7B1%7D%7B8%7D+" id="TexFormula13" title="\displaystyle 16)\,\, \frac{(3\cdot 2^{20}+7\cdot 2^{19})\cdot 52}{(-1)^7\cdot(13\cdot8^4)^2} = -\frac{2^{21}(3\cdot 2+7)\cdot 13}{13^2\cdot2^{24}} = -\frac{13^2}{13^2\cdot 2^3} =- \frac{1}{8}

аноним 06 Апр, 18