Упр.384 ГДЗ Никольский Потапов 9 класс (Алгебра)
Вычислите:
а) $$\left(\frac{1}{2}\right)^{\frac{1}{2}}\cdot 8^{\frac{1}{3}}\cdot 2^{-\frac{1}{2}}$$
б) $$\left(\frac{1}{27}\right)^{-\frac{1}{9}}\cdot 6^{\frac{1}{2}}\cdot 2^{-\frac{1}{2}}\cdot 3^{\frac{1}{6}}$$
в) $$\left(3^{1{,}5}-2^{1{,}5}\right)\left(3^{1{,}5}+2^{1{,}5}\right)$$
г) $$\left(2^{2{,}5}-3^{1{,}5}\right)\left(2^{2{,}5}+3^{1{,}5}\right)$$
д) $$\left(5^{\frac{1}{3}}-2^{\frac{2}{3}}\right)\left(5^{\frac{2}{3}}+2^{\frac{2}{3}}\cdot 5^{\frac{1}{3}}+4^{\frac{2}{3}}\right)$$
е) $$\left(\left(3^{\frac{2}{3}}+7^{\frac{1}{3}}\right)\left(9^{\frac{2}{3}}-3^{\frac{2}{3}}\cdot 7^{\frac{1}{3}}+49^{\frac{1}{3}}\right)\right)^{\frac{3}{4}}$$
а)
$$\left(\frac12\right)^{\frac12}\cdot 8^{\frac13}\cdot 2^{-\frac12} =2^{-\frac12}\cdot (2^3)^{\frac13}\cdot 2^{-\frac12} =2^{-\frac12}\cdot 2\cdot 2^{-\frac12} =2^{0}=1.$$
б)
$$\left(\frac1{27}\right)^{-\frac19}\cdot 6^{\frac12}\cdot 2^{-\frac12}\cdot 3^{\frac16} =(3^{-3})^{-\frac19}\cdot (2\cdot 3)^{\frac12}\cdot 2^{-\frac12}\cdot 3^{\frac16}$$
$$=3^{\frac13}\cdot 2^{\frac12}\cdot 3^{\frac12}\cdot 2^{-\frac12}\cdot 3^{\frac16} =3^{\frac13+\frac12+\frac16} =3^1=3.$$в)
$$\left(3^{1,5}-2^{1,5}\right)\left(3^{1,5}+2^{1,5}\right) =\left(3^{1,5}\right)^2-\left(2^{1,5}\right)^2 =3^3-2^3=27-8=19.$$
г)
$$\left(2^{2,5}-3^{1,5}\right)\left(2^{2,5}+3^{1,5}\right) =\left(2^{2,5}\right)^2-\left(3^{1,5}\right)^2 =2^5-3^3=32-27=5.$$
д)
$$\left(5^{\frac13}-2^{\frac23}\right)\left(5^{\frac23}+2^{\frac23}\cdot 5^{\frac13}+4^{\frac23}\right)$$
$$=\left(5^{\frac13}\right)^3-\left(2^{\frac23}\right)^3 =5-4=1.$$е)
$$\left(\left(3^{\frac23}+7^{\frac13}\right)\left(9^{\frac23}-3^{\frac23}\cdot 7^{\frac13}+49^{\frac13}\right)\right)^{\frac34}$$
$$=\left(\left(3^{\frac23}+7^{\frac13}\right)\left(\left(3^{\frac23}\right)^2-3^{\frac23}\cdot 7^{\frac13}+\left(7^{\frac13}\right)^2\right)\right)^{\frac34}$$
$$=\left(\left(3^{\frac23}\right)^3+\left(7^{\frac13}\right)^3\right)^{\frac34} =\left(3^2+7\right)^{\frac34} =16^{\frac34} =\left(2^4\right)^{\frac34} =2^3=8.$$
Ответ
а) 1; б) 3; в) 19; г) 5; д) 1; е) 8.
