Упр.23.21 ГДЗ Мерзляк Поляков 9 класс (Алгебра)

1) 12^(1/3)·6^(2/3)·(0,5)^(1/3);
2) 25^1,5+(0,25)^(-0,5)-81^0,75;
3) (1/16)^(-3/4)+(1/8)^(-2/3)·(0,81)^(-0,5);
4) 16^(1/8)·8^(-5/6)·4^1,5;
5) (10 000^0,4·10^0,5)/(100^0,3·1000^(1/6));
6) (5^(3/2)·8^(1/12))/9^(1/6)·8^(1/4)/(5^(5/2)·9^(1/3));
7) (72^(2/3))^(1/2)·2^(-4/3):36^(-1/6);
8) ((3^(-5/6)·7^(-5/6))/(21^(-1)·5^(1/3)))^(-6).

1. $$12^{\frac13}\cdot 6^{\frac23}\cdot (0{,}5)^{\frac13}=(6\cdot 2)^{\frac13}\cdot 6^{\frac23}\cdot \left(\frac12\right)^{\frac13}$$
   $$=6^{\frac13}\cdot 2^{\frac13}\cdot 6^{\frac23}\cdot 2^{-\frac13}=6^{\frac13+\frac23}\cdot 2^{\frac13-\frac13}=6\cdot 1=6.$$

2. $$25^{1{,}5}+(0{,}25)^{-0{,}5}-81^{0{,}75}=(5^2)^{\frac32}+\left(\frac14\right)^{-\frac12}-(3^4)^{\frac34}$$
   $$=5^3+4^{\frac12}-3^3=125+2-27=100.$$

3. $$\left(\frac1{16}\right)^{-\frac34}+\left(\frac18\right)^{-\frac23}\cdot (0{,}81)^{-0{,}5}=16^{\frac34}+8^{\frac23}\cdot \left(\frac{81}{100}\right)^{-\frac12}$$
   $$=(2^4)^{\frac34}+(2^3)^{\frac23}\cdot \left(\frac{100}{81}\right)^{\frac12}=2^3+2^2\cdot \frac{10}{9}$$
   $$=8+\frac{40}{9}=12\frac49.$$

4. $$16^{\frac18}\cdot 8^{-\frac56}\cdot 4^{1{,}5}=(2^4)^{\frac18}\cdot (2^3)^{-\frac56}\cdot (2^2)^{\frac32}$$
   $$=2^{\frac12}\cdot 2^{-\frac52}\cdot 2^3=2^{\frac12-\frac52+3}=2^2=4.$$

5. $$\frac{10\,000^{0{,}4}\cdot 10^{0{,}5}}{100^{0{,}3}\cdot 1000^{\frac16}}=\frac{(10^4)^{0{,}4}\cdot 10^{0{,}5}}{(10^2)^{0{,}3}\cdot (10^3)^{\frac16}}$$
   $$=\frac{10^{1{,}6}\cdot 10^{0{,}5}}{10^{0{,}6}\cdot 10^{0{,}5}}=10^{1{,}6-0{,}6}=10.$$

6. $$\frac{5^{\frac32}\cdot 8^{\frac1{12}}}{9^{\frac16}}\cdot \frac{8^{\frac14}}{5^{\frac52}\cdot 9^{\frac13}}= \frac{5^{\frac32-\frac52}\cdot 8^{\frac1{12}+\frac14}}{9^{\frac16+\frac13}}$$
   $$=\frac{5^{-1}\cdot 8^{\frac13}}{9^{\frac12}}=\frac{5^{-1}\cdot 2}{3}=\frac{2}{15}.$$

7. $$\left(72^{\frac23}\right)^{\frac12}\cdot 2^{-\frac43}:36^{-\frac16}=72^{\frac13}\cdot 2^{-\frac43}\cdot 36^{\frac16}$$
   $$=(36\cdot 2)^{\frac13}\cdot 2^{-\frac43}\cdot 36^{\frac16}=36^{\frac13}\cdot 2^{\frac13-\frac43}\cdot 36^{\frac16}$$
   $$=36^{\frac12}\cdot 2^{-1}=6\cdot \frac12=3.$$

8. $$\left(\frac{3^{-\frac56}\cdot 7^{-\frac56}}{21^{-1}\cdot 5^{\frac13}}\right)^{-6} =\left(\frac{21^{-1}\cdot 5^{\frac13}}{3^{-\frac56}\cdot 7^{-\frac56}}\right)^6$$
   $$=\left(\frac{3^{\frac56}\cdot 7^{\frac56}\cdot 5^{\frac13}}{21}\right)^6 =\left(\frac{21^{\frac56}\cdot 5^{\frac13}}{21}\right)^6$$
   $$=\left(21^{-\frac16}\cdot 5^{\frac13}\right)^6 =21^{-1}\cdot 5^2=\frac{25}{21}=1\frac4{21}.$$

Ответ

1) $$6$$; 2) $$100$$; 3) $$12\frac49$$; 4) $$4$$; 5) $$10$$; 6) $$\frac{2}{15}$$; 7) $$3$$; 8) $$1\frac4{21}$$.