Задание 28
1) $\sqrt[6]{36^3} = (36^3)^{\frac{1}{6}} = 36^{\frac{3}{6}} = 36^{\frac{1}{2}} = \sqrt{36} = 6$
2) $\sqrt[12]{64^2} = (64^2)^{\frac{1}{12}} = 64^{\frac{2}{12}} = 64^{\frac{1}{6}} = \sqrt[6]{64} = 2$
3) $\sqrt[4]{\sqrt{\frac{1}{25}}} = \sqrt[4]{\frac{1}{5}} = (\frac{1}{5})^{\frac{1}{4}} $
4) $\sqrt[8]{225^4} = (225^4)^{\frac{1}{8}} = 225^{\frac{4}{8}} = 225^{\frac{1}{2}} = \sqrt{225} = 15$
Задание 29
1) $\sqrt[3]{10^6} = (10^6)^{\frac{1}{3}} = 10^{\frac{6}{3}} = 10^2 = 100$
2) $\sqrt[3]{3^{12}} = (3^{12})^{\frac{1}{3}} = 3^{\frac{12}{3}} = 3^4 = 81$
3) $\sqrt[4]{(\frac{1}{2})^{12}} = ((\frac{1}{2})^{12})^{\frac{1}{4}} = (\frac{1}{2})^{\frac{12}{4}} = (\frac{1}{2})^3 = \frac{1}{8}$
4) $\sqrt[4]{(\frac{1}{3})^{16}} = ((\frac{1}{3})^{16})^{\frac{1}{4}} = (\frac{1}{3})^{\frac{16}{4}} = (\frac{1}{3})^4 = \frac{1}{81}$
Задание 30
1) $\sqrt[3]{-8} = -2$, так как $(-2)^3 = -8$
2) $\sqrt[15]{-1} = -1$, так как $(-1)^{15} = -1$
3) $\sqrt[3]{\frac{1}{27}} = \frac{1}{3}$, так как $(\frac{1}{3})^3 = \frac{1}{27}$
4) $\sqrt[5]{-1024} = -4$, так как $(-4)^5 = -1024$
5) $\sqrt[3]{-343} = -7$, так как $(-7)^3 = -343$
6) $\sqrt[7]{-8^7} = -8$
Задание 31
1) $x^4 = 256$
\$x = \pm \sqrt[4]{256} = \pm 4$
2) $x^5 = -\frac{1}{32}$
\$x = \sqrt[5]{-\frac{1}{32}} = -\frac{1}{2}$
3) $5x^5 = -160$
\$x^5 = -32$
\$x = \sqrt[5]{-32} = -2$
4) $2x^6 = 128$
\$x^6 = 64$
\$x = \pm \sqrt[6]{64} = \pm 2$
Задание 32
1) $\sqrt[3]{-125} + \frac{1}{8} \sqrt[6]{64} = -5 + \frac{1}{8} \cdot 2 = -5 + \frac{1}{4} = -4\frac{3}{4} = -4,75$
2) $\sqrt[5]{32} - 0,5 \sqrt[3]{-216} = 2 - 0,5 \cdot (-6) = 2 + 3 = 5$
3) $- \frac{1}{3} \sqrt[4]{81} + \sqrt{625} = - \frac{1}{3} \cdot 3 + 25 = -1 + 25 = 24$
4) $\sqrt[3]{-1000} - \frac{1}{4} \sqrt[4]{256} = -10 - \frac{1}{4} \cdot 4 = -10 - 1 = -11$
5) $\sqrt[5]{\frac{1}{243}} + \sqrt[3]{-0,001} - \sqrt[4]{0,0016} = \frac{1}{3} + (-0,1) - 0,2 = \frac{1}{3} - 0,3 = \frac{1}{3} - \frac{3}{10} = \frac{10 - 9}{30} = \frac{1}{30}$
Задание 33
1) $\sqrt[3]{343 \cdot 0,125} = \sqrt[3]{343} \cdot \sqrt[3]{0,125} = 7 \cdot 0,5 = 3,5$
2) $\sqrt[3]{512 \cdot 216} = \sqrt[3]{512} \cdot \sqrt[3]{216} = 8 \cdot 6 = 48$
3) $\sqrt[5]{32 \cdot 100000} = \sqrt[5]{32} \cdot \sqrt[5]{100000} = 2 \cdot 10 = 20$
Задание 34
1) $\sqrt[4]{5^3 \cdot 7^3} = \sqrt[4]{125 \cdot 343} = \sqrt[4]{42875}$
2) $\sqrt[4]{11^4 \cdot 3^4} = \sqrt[4]{11^4} \cdot \sqrt[4]{3^4} = 11 \cdot 3 = 33$
3) $\sqrt[5]{(0,2)^5 \cdot 8^5} = \sqrt[5]{(0,2)^5} \cdot \sqrt[5]{8^5} = 0,2 \cdot 8 = 1,6$
4) $\sqrt[7]{(\frac{1}{3})^7 \cdot 21^7} = \sqrt[7]{(\frac{1}{3})^7} \cdot \sqrt[7]{21^7} = \frac{1}{3} \cdot 21 = 7$
Задание 35
1) $\sqrt[3]{2} \cdot \sqrt[3]{500} = \sqrt[3]{2 \cdot 500} = \sqrt[3]{1000} = 10$
2) $\sqrt[3]{0,2} \cdot \sqrt[3]{0,04} = \sqrt[3]{0,2 \cdot 0,04} = \sqrt[3]{0,008} = 0,2$
3) $\sqrt[3]{324} \cdot \sqrt[3]{4} = \sqrt[3]{324 \cdot 4} = \sqrt[3]{1296}$
4) $\sqrt[5]{2} \cdot \sqrt[5]{16} = \sqrt[5]{2 \cdot 16} = \sqrt[5]{32} = 2$
Задание 36
1) $\sqrt[5]{3^{10} \cdot 2^{15}} = \sqrt[5]{3^{10}} \cdot \sqrt[5]{2^{15}} = 3^{\frac{10}{5}} \cdot 2^{\frac{15}{5}} = 3^2 \cdot 2^3 = 9 \cdot 8 = 72$
2) $\sqrt[3]{2^3 \cdot 5^6} = \sqrt[3]{2^3} \cdot \sqrt[3]{5^6} = 2 \cdot 5^{\frac{6}{3}} = 2 \cdot 5^2 = 2 \cdot 25 = 50$
3) $\sqrt[4]{3^{12} \cdot (\frac{1}{3})^8} = \sqrt[4]{3^{12}} \cdot \sqrt[4]{(\frac{1}{3})^8} = 3^{\frac{12}{4}} \cdot (\frac{1}{3})^{\frac{8}{4}} = 3^3 \cdot (\frac{1}{3})^2 = 27 \cdot \frac{1}{9} = 3$
4) $\sqrt[10]{4^{30} \cdot (\frac{1}{2})^{20}} = \sqrt[10]{4^{30}} \cdot \sqrt[10]{(\frac{1}{2})^{20}} = 4^{\frac{30}{10}} \cdot (\frac{1}{2})^{\frac{20}{10}} = 4^3 \cdot (\frac{1}{2})^2 = 64 \cdot \frac{1}{4} = 16$
