10. Возведение в степень произведения и степени. A (ab) (xy) (2y)5 (ab) (3.10)4 (-2x3y2)5 (64)3 (xyz2)6 (2x3)4 (m5) ((x2)4)7 (p3)5: p10 (y2)7y3 (α4)5: (α³)6 b13: (b2)6

Давай разберем по порядку, как решать задания на возведение в степень произведения и степени.

A

1. \[(ab)^7 = a^7b^7\]
2. \[(xy)^7 = x^7y^7\]
3. \[(2y)^5 = 2^5y^5 = 32y^5\]
4. \[(\frac{1}{3}ab)^3 = \frac{1}{3^3}a^3b^3 = \frac{1}{27}a^3b^3\]
5. \[(3 \cdot 10)^4 = 3^4 \cdot 10^4 = 81 \cdot 10000 = 810000\]
6. \[(-2x^3y^2)^5 = (-2)^5(x^3)^5(y^2)^5 = -32x^{15}y^{10}\]
7. \[(b^4)^3 = b^{4 \cdot 3} = b^{12}\]
8. \[(xyz^2)^6 = x^6y^6(z^2)^6 = x^6y^6z^{12}\]
9. \[(2x^3)^4 = 2^4(x^3)^4 = 16x^{12}\]
10. \[(m^5)^n = m^{5n}\]
11. \[((x^2)^4)^7 = (x^{2 \cdot 4})^7 = (x^8)^7 = x^{8 \cdot 7} = x^{56}\]
12. \[(p^3)^5 : p^{10} = p^{3 \cdot 5} : p^{10} = p^{15} : p^{10} = p^{15-10} = p^5\]
13. \[(y^2)^7 \cdot y^3 = y^{2 \cdot 7} \cdot y^3 = y^{14} \cdot y^3 = y^{14+3} = y^{17}\]
14. \[(\alpha^4)^5 : (\alpha^3)^6 = \alpha^{4 \cdot 5} : \alpha^{3 \cdot 6} = \alpha^{20} : \alpha^{18} = \alpha^{20-18} = \alpha^2\]
15. \[b^{13} : (b^2)^6 = b^{13} : b^{2 \cdot 6} = b^{13} : b^{12} = b^{13-12} = b^1 = b\]

B

1. \[(3x^2y^4)^3 = 3^3(x^2)^3(y^4)^3 = 27x^6y^{12}\]
2. \[(-2x^3y^5)^6 = (-2)^6(x^3)^6(y^5)^6 = 64x^{18}y^{30}\]
3. \[b^{19} : (b^3b^2)^3 = b^{19} : (b^{3+2})^3 = b^{19} : (b^5)^3 = b^{19} : b^{5 \cdot 3} = b^{19} : b^{15} = b^{19-15} = b^4\]
4. \[(\frac{p^2}{4a})^3 = \frac{(p^2)^3}{4^3a^3} = \frac{p^{2 \cdot 3}}{64a^3} = \frac{p^6}{64a^3}\]
5. \[(\frac{\alpha^2 \cdot b^3}{2p})^2 = \frac{(\alpha^2)^2 \cdot (b^3)^2}{(2p)^2} = \frac{\alpha^{2 \cdot 2} \cdot b^{3 \cdot 2}}{4p^2} = \frac{\alpha^4b^6}{4p^2}\]
6. \[(-\frac{1}{3}\alpha^4b^5)^3 = (-\frac{1}{3})^3(\alpha^4)^3(b^5)^3 = -\frac{1}{27}\alpha^{12}b^{15}\]
7. \[(-2\frac{1}{3}x^5y^3)^2 = (- \frac{7}{3}x^5y^3)^2 = \frac{49}{9}x^{10}y^6\]
8. \[(2\frac{1}{2}n^6m^4)^4 = (\frac{5}{2}n^6m^4)^4 = (\frac{5}{2})^4(n^6)^4(m^4)^4 = \frac{625}{16}n^{24}m^{16}\]
9. \[(\frac{7^9 \cdot 7^2}{7^{10}})^2 = (\frac{7^{9+2}}{7^{10}})^2 = (\frac{7^{11}}{7^{10}})^2 = (7^{11-10})^2 = (7^1)^2 = 7^2 = 49\]
10. \[(-2x^6y^4)^3 = (-2)^3(x^6)^3(y^4)^3 = -8x^{18}y^{12}\]
11. \[\alpha^{18} : (\alpha^3)^5 \cdot \alpha^0 = \alpha^{18} : \alpha^{3 \cdot 5} \cdot 1 = \alpha^{18} : \alpha^{15} = \alpha^{18-15} = \alpha^3\]
12. \[(b^{10} \cdot b^2)^3 : b^{20} = (b^{10+2})^3 : b^{20} = (b^{12})^3 : b^{20} = b^{12 \cdot 3} : b^{20} = b^{36} : b^{20} = b^{36-20} = b^{16}\]
13. \[(\alpha^6)^2 : (\alpha^2)^4 \cdot \alpha^5 = \alpha^{6 \cdot 2} : \alpha^{2 \cdot 4} \cdot \alpha^5 = \alpha^{12} : \alpha^8 \cdot \alpha^5 = \alpha^{12-8} \cdot \alpha^5 = \alpha^4 \cdot \alpha^5 = \alpha^{4+5} = \alpha^9\]
14. \[((p^2)^3)^5 = (p^{2 \cdot 3})^5 = (p^6)^5 = p^{6 \cdot 5} = p^{30}\]
15. \[(x^3)^8 : (x^4)^6 = x^{3 \cdot 8} : x^{4 \cdot 6} = x^{24} : x^{24} = 1\]

C

1. \[(\alpha^{m+1})^2 : (\alpha^{m-1})^2 = \alpha^{2(m+1)} : \alpha^{2(m-1)} = \alpha^{2m+2} : \alpha^{2m-2} = \alpha^{2m+2-(2m-2)} = \alpha^{2m+2-2m+2} = \alpha^4\]
2. \[(c^{n+1})^4 : (c^{n-2})^3 = c^{4(n+1)} : c^{3(n-2)} = c^{4n+4} : c^{3n-6} = c^{4n+4-(3n-6)} = c^{4n+4-3n+6} = c^{n+10}\]
3. \[(b^{3m})^2 : (b^{2m-1})^3 = b^{3m \cdot 2} : b^{3(2m-1)} = b^{6m} : b^{6m-3} = b^{6m-(6m-3)} = b^{6m-6m+3} = b^3\]
4. \[(-3\frac{1}{3}\alpha^2b^6)^2 = (-\frac{10}{3}\alpha^2b^6)^2 = \frac{100}{9}\alpha^4b^{12}\]
5. \[(-\frac{1}{2}n^5m^3)^4 = (\frac{1}{2})^4(n^5)^4(m^3)^4 = \frac{1}{16}n^{20}m^{12}\]
6. \[\frac{5^6 \cdot 25^2}{125^3} = \frac{5^6 \cdot (5^2)^2}{(5^3)^3} = \frac{5^6 \cdot 5^4}{5^9} = \frac{5^{6+4}}{5^9} = \frac{5^{10}}{5^9} = 5^{10-9} = 5^1 = 5\]
7. \[\frac{27^4 \cdot 3^2}{81^3} = \frac{(3^3)^4 \cdot 3^2}{(3^4)^3} = \frac{3^{12} \cdot 3^2}{3^{12}} = \frac{3^{12+2}}{3^{12}} = \frac{3^{14}}{3^{12}} = 3^{14-12} = 3^2 = 9\]
8. \[\frac{16^3 \cdot 8^2}{64^3} = \frac{(2^4)^3 \cdot (2^3)^2}{(2^6)^3} = \frac{2^{12} \cdot 2^6}{2^{18}} = \frac{2^{12+6}}{2^{18}} = \frac{2^{18}}{2^{18}} = 1\]
9. \[\frac{64^2 \cdot 4^3}{16^4} = \frac{(4^3)^2 \cdot 4^3}{(4^2)^4} = \frac{4^6 \cdot 4^3}{4^8} = \frac{4^{6+3}}{4^8} = \frac{4^9}{4^8} = 4^{9-8} = 4\]
10. \[2^8 \cdot (2^3)^2 : 2^{10} = 2^8 \cdot 2^{3 \cdot 2} : 2^{10} = 2^8 \cdot 2^6 : 2^{10} = 2^{8+6} : 2^{10} = 2^{14} : 2^{10} = 2^{14-10} = 2^4 = 16\]
11. \[6^{12} : (6^5)^2 \cdot 6^0 = 6^{12} : 6^{5 \cdot 2} \cdot 1 = 6^{12} : 6^{10} = 6^{12-10} = 6^2 = 36\]
12. \[(c^{4n+1})^3 : (c^{6n-2})^2 = c^{3(4n+1)} : c^{2(6n-2)} = c^{12n+3} : c^{12n-4} = c^{12n+3-(12n-4)} = c^{12n+3-12n+4} = c^7\]
13. \[(x^{n-3} \cdot x^{n+2})^2 = (x^{n-3+n+2})^2 = (x^{2n-1})^2 = x^{2(2n-1)} = x^{4n-2}\]
14. \[(\alpha^{m+1})^2 : \alpha^{m-1} = \alpha^{2(m+1)} : \alpha^{m-1} = \alpha^{2m+2} : \alpha^{m-1} = \alpha^{2m+2-(m-1)} = \alpha^{2m+2-m+1} = \alpha^{m+3}\]
15. \[\alpha^{5n+3} : (\alpha^n)^4 = \alpha^{5n+3} : \alpha^{4n} = \alpha^{5n+3-4n} = \alpha^{n+3}\]

Ответ: Решения выше.