Упр.384 ГДЗ Никольский Потапов 9 класс (Алгебра)

а) (1/2)^(1/2)•8^(1/3)•2^(-1/2)
б) (1/27)^(-1/9)•6^(1/2)•2^(-1/2)•3^(1/6)
в) (3^1,5-2^1,5 )(3^1,5+2^1,5 )
г) (2^2,5-3^1,5 )(2^2,5+3^1,5 )
д) (5^(1/3)-2^(2/3))(5^(2/3)+2^(2/3)•5^(1/3)+4^(2/3))
е) ((3^(2/3)+7^(1/3))(9^(2/3)-3^(2/3)•7^(1/3)+49^(1/3)))^(3/4)

1. а)

   $$\left(\frac12\right)^{\frac12}\cdot 8^{\frac13}\cdot 2^{-\frac12} =2^{-\frac12}\cdot (2^3)^{\frac13}\cdot 2^{-\frac12} =2^{-\frac12}\cdot 2\cdot 2^{-\frac12} =2^{0}=1.$$

2. б)

   $$\left(\frac1{27}\right)^{-\frac19}\cdot 6^{\frac12}\cdot 2^{-\frac12}\cdot 3^{\frac16} =(3^{-3})^{-\frac19}\cdot (2\cdot 3)^{\frac12}\cdot 2^{-\frac12}\cdot 3^{\frac16}$$
   $$=3^{\frac13}\cdot 2^{\frac12}\cdot 3^{\frac12}\cdot 2^{-\frac12}\cdot 3^{\frac16} =3^{\frac13+\frac12+\frac16} =3^1=3.$$

3. в)

   $$\left(3^{1,5}-2^{1,5}\right)\left(3^{1,5}+2^{1,5}\right) =\left(3^{1,5}\right)^2-\left(2^{1,5}\right)^2 =3^3-2^3=27-8=19.$$

4. г)

   $$\left(2^{2,5}-3^{1,5}\right)\left(2^{2,5}+3^{1,5}\right) =\left(2^{2,5}\right)^2-\left(3^{1,5}\right)^2 =2^5-3^3=32-27=5.$$

5. д)

   $$\left(5^{\frac13}-2^{\frac23}\right)\left(5^{\frac23}+2^{\frac23}\cdot 5^{\frac13}+4^{\frac23}\right)$$
   $$=\left(5^{\frac13}\right)^3-\left(2^{\frac23}\right)^3 =5-4=1.$$

6. 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.