Упр.13.16 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

1) (343^(1/2)·(1/49)^(3/8))^(4/3);
2) 10^(1/4)·40^(1/4)·5^(1/2);
3) (32^0,24·4^0,7)/(64^0,6·16^0,25);
4O) (12^(1/2)/(7^(2/3)·8^(-1/6))·(3^(1/2)·7^(5/3))/8^(1/2).

1. $$\left(343^{\frac12}\cdot\left(\frac1{49}\right)^{\frac38}\right)^{\frac43}$$
   $$=\left((7^3)^{\frac12}\cdot(7^{-2})^{\frac38}\right)^{\frac43}$$
   $$=\left(7^{\frac32}\cdot 7^{-\frac34}\right)^{\frac43}=\left(7^{\frac32-\frac34}\right)^{\frac43}=\left(7^{\frac34}\right)^{\frac43}$$
   $$=7^{\frac34\cdot\frac43}=7.$$
2. $$10^{\frac14}\cdot 40^{\frac14}\cdot 5^{\frac12}$$
   $$=(5\cdot 2)^{\frac14}\cdot(5\cdot 2^3)^{\frac14}\cdot 5^{\frac12}$$
   $$=5^{\frac14}\cdot 2^{\frac14}\cdot 5^{\frac14}\cdot 2^{\frac34}\cdot 5^{\frac12}$$
   $$=5^{\frac14+\frac14+\frac12}\cdot 2^{\frac14+\frac34}=5^1\cdot 2^1=10.$$
3. $$\frac{32^{0{,}24}\cdot 4^{0{,}7}}{64^{0{,}6}\cdot 16^{0{,}25}}$$
   $$=\frac{(2^5)^{0{,}24}\cdot(2^2)^{0{,}7}}{(2^6)^{0{,}6}\cdot(2^4)^{0{,}25}}$$
   $$=\frac{2^{1{,}2}\cdot 2^{1{,}4}}{2^{3{,}6}\cdot 2^1} =\frac{2^{2{,}6}}{2^{4{,}6}}=2^{-2}=\frac14.$$
4. $$\frac{12^{\frac12}}{7^{\frac23}\cdot 8^{-\frac16}}\cdot \frac{3^{\frac12}\cdot 7^{\frac53}}{8^{\frac12}}$$
   $$=\frac{(3\cdot 4)^{\frac12}}{7^{\frac23}\cdot (2^3)^{-\frac16}}\cdot \frac{3^{\frac12}\cdot 7^{\frac53}}{(2^3)^{\frac12}}$$
   $$=\frac{3^{\frac12}\cdot 2}{7^{\frac23}\cdot 2^{-\frac12}}\cdot \frac{3^{\frac12}\cdot 7^{\frac53}}{2^{\frac32}}$$
   $$=\frac{3^{\frac12}\cdot 2^{\frac32}}{7^{\frac23}}\cdot \frac{3^{\frac12}\cdot 7^{\frac53}}{2^{\frac32}}$$
   $$=3^{\frac12+\frac12}\cdot 7^{\frac53-\frac23}=3\cdot 7=21.$$

Ответ

1) $$7$$; 2) $$10$$; 3) $$\frac14$$; 4) $$21$$.