Упр.13.15 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

1) (1/16)^(-3/4)+(1/8)^(-2/3)·(0,81)^(-0,5);
2) 16^(1/8)·8^(-5/6)·4^1,5;
3) (5^(3/2)·8^(1/12))/9^(1/6)·8^(1/4)/(5^(5/2)·9^(1/3));
4) (72^(2/3))^(1/2)·2^(-4/3):36^(-1/6).

1. $$\left(\frac{1}{16}\right)^{-\frac{3}{4}}+\left(\frac{1}{8}\right)^{-\frac{2}{3}}\cdot(0{,}81)^{-0{,}5}$$

   $$\left(\frac{1}{2^4}\right)^{-\frac{3}{4}}+\left(\frac{1}{2^3}\right)^{-\frac{2}{3}}\cdot(0{,}9^2)^{-0{,}5}$$

   $$\left(\frac{1}{2}\right)^{-3}+\left(\frac{1}{2}\right)^{-2}\cdot(0{,}9)^{-1}$$

   $$8+4\cdot\frac{10}{9}=8+\frac{40}{9}=8+4\frac{4}{9}=12\frac{4}{9}$$

2. $$16^{\frac{1}{8}}\cdot 8^{-\frac{5}{6}}\cdot 4^{1{,}5}$$

   $$\left(2^4\right)^{\frac{1}{8}}\cdot\left(2^3\right)^{-\frac{5}{6}}\cdot\left(2^2\right)^{1{,}5}$$

   $$2^{\frac{1}{2}}\cdot 2^{-\frac{5}{2}}\cdot 2^3=2^{\frac{1}{2}-\frac{5}{2}+3}=2^2=2$$

3. $$\frac{5^{\frac{3}{2}}\cdot 8^{\frac{1}{12}}}{9^{\frac{1}{6}}}\cdot\frac{8^{\frac{1}{4}}}{5^{\frac{5}{2}}\cdot 9^{\frac{1}{3}}}$$

   $$\frac{5^{\frac{3}{2}}\cdot 8^{\frac{1}{12}}\cdot 8^{\frac{1}{4}}}{5^{\frac{5}{2}}\cdot 9^{\frac{1}{6}}\cdot 9^{\frac{1}{3}}} =\frac{5^{\frac{3}{2}}\cdot 8^{\frac{1}{3}}}{5^{\frac{5}{2}}\cdot 9^{\frac{1}{2}}}$$

   $$\frac{5^{\frac{3}{2}}}{5^{\frac{5}{2}}}\cdot\frac{8^{\frac{1}{3}}}{9^{\frac{1}{2}}} =\frac{1}{5}\cdot\frac{2}{3}=\frac{2}{15}$$

4. $$(72^{\frac{2}{3}})^{\frac{1}{2}}\cdot 2^{-\frac{4}{3}}:36^{-\frac{1}{6}}$$

   $$72^{\frac{1}{3}}\cdot 2^{-\frac{4}{3}}\cdot 36^{\frac{1}{6}}$$

   $$\left(8\cdot 9\right)^{\frac{1}{3}}\cdot 2^{-\frac{4}{3}}\cdot\left(9\cdot 4\right)^{\frac{1}{6}} =\left(2^3\cdot 3^2\right)^{\frac{1}{3}}\cdot 2^{-\frac{4}{3}}\cdot\left(3^2\cdot 2^2\right)^{\frac{1}{6}}$$

   $$2\cdot 3^{\frac{2}{3}}\cdot 2^{-\frac{4}{3}}\cdot 3^{\frac{1}{3}}\cdot 2^{\frac{1}{3}} =2^{1-\frac{4}{3}+\frac{1}{3}}\cdot 3^{\frac{2}{3}+\frac{1}{3}}=2^0\cdot 3=3$$

Ответ

1) $$12\frac{4}{9}$$; 2) $$2$$; 3) $$\frac{2}{15}$$; 4) $$3$$.