10. Возведение в степень произведения и степени. A (ab) (xy) (2y)5 (ab) (3.10)4 (-2x3y2)5 (b4)3 (xyz2)6 (2x3)4 (m5) ((x2)4)7 (p3)5: p10 (y2)7.y3 (α4)5: (α3)6 b13: (b2)6 B (3x²y4)3 (-2x3y5)6 b19: (b3b2)3 (4α) (α2.b3)2 2p (-α4b5)3 (-2x5y3 (2nm4) ( 710 (-2x6y4)3 α18: (α3)5. α (b10.b2)3: b20 (α6)2 : (α2)4. α5 ((p2)3)5 (x3)8: (x4)6 C (am+1)2: (am-1)2 (cn+1)4: (cn-2)3 (b3m)2: (b2m-1)3 (-3α2b6)2 (-n5m3)4 56.252 1253 274-32 813 163.82 643 642.43 4 ( 164 28. (23)2: 210 612: (65)2.60 (c4n+1)3: (c6n-2)2 (x-3x+2)2 (am+1)2: am-1 a5n+3: (α)4

Решение задач на степени произведения и степени

Давай разберем задачи на возведение в степень произведения и степени.

Столбец A

1. \[ (ab)^7 = a^7b^7 \]

2. \[ (xy)^7 = x^7y^7 \]

3. \[ (2y)^5 = 2^5 y^5 = 32y^5 \]

4. \[ \left(\frac{1}{3}ab\right)^3 = \frac{1}{3^3}a^3b^3 = \frac{1}{27}a^3b^3 \]

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. \[ (xyz^2)^6 = x^6 y^6 (z^2)^6 = x^6y^6z^{12} \]

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. \[ b^{13} : (b^2)^6 = b^{13} : b^{2 \cdot 6} = b^{13} : b^{12} = b^{13-12} = b \]

Столбец 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. \[ b^{19} : (b^3b^2)^3 = b^{19} : (b^{3+2})^3 = b^{19} : (b^5)^3 = b^{19} : b^{15} = b^{19-15} = b^4 \]

4. \[ \left(\frac{p^2}{4\alpha}\right)^3 = \frac{(p^2)^3}{(4\alpha)^3} = \frac{p^6}{4^3 \alpha^3} = \frac{p^6}{64\alpha^3} \]

5. \[ \left(\frac{\alpha^2 \cdot b^3}{2p}\right)^2 = \frac{(\alpha^2)^2 (b^3)^2}{(2p)^2} = \frac{\alpha^4 b^6}{4p^2} \]

6. \[ \left(-\frac{1}{3}\alpha^4b^5\right)^3 = \left(-\frac{1}{3}\right)^3 (\alpha^4)^3 (b^5)^3 = -\frac{1}{27}\alpha^{12}b^{15} \]

7. \[ \left(-\frac{2}{3}x^5y^3\right)^2 = \left(-\frac{2}{3}\right)^2 (x^5)^2 (y^3)^2 = \frac{4}{9}x^{10}y^6 \]

8. \[ \left(2\frac{1}{2}n^6m^4\right)^4 = \left(\frac{5}{2}n^6m^4\right)^4 = \left(\frac{5}{2}\right)^4 (n^6)^4 (m^4)^4 = \frac{625}{16}n^{24}m^{16} \]

9. \[ \left(\frac{7^9 \cdot 7^2}{7^{10}}\right)^2 = \left(\frac{7^{9+2}}{7^{10}}\right)^2 = \left(\frac{7^{11}}{7^{10}}\right)^2 = (7^{11-10})^2 = (7^1)^2 = 7^2 = 49 \]

10. \[ (-2x^6y^4)^3 = (-2)^3 (x^6)^3 (y^4)^3 = -8x^{18}y^{12} \]

11. \[ \alpha^{18} : (\alpha^3)^5 \cdot \alpha^0 = \alpha^{18} : \alpha^{15} \cdot 1 = \alpha^{18-15} = \alpha^3 \]

12. \[ (b^{10} \cdot b^2)^3 : b^{20} = (b^{10+2})^3 : b^{20} = (b^{12})^3 : b^{20} = b^{36} : b^{20} = b^{36-20} = b^{16} \]

13. \[ (\alpha^6)^2 : (\alpha^2)^4 \cdot \alpha^5 = \alpha^{12} : \alpha^8 \cdot \alpha^5 = \alpha^{12-8} \cdot \alpha^5 = \alpha^4 \cdot \alpha^5 = \alpha^{4+5} = \alpha^9 \]

14. \[ ((p^2)^3)^5 = (p^{2 \cdot 3})^5 = (p^6)^5 = p^{6 \cdot 5} = p^{30} \]

15. \[ (x^3)^8 : (x^4)^6 = x^{24} : x^{24} = 1 \]

Столбец C

1. \[ (a^{m+1})^2 : (a^{m-1})^2 = a^{2(m+1)} : a^{2(m-1)} = a^{2m+2} : a^{2m-2} = a^{(2m+2)-(2m-2)} = a^{2m+2-2m+2} = a^4 \]

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. \[ (b^{3m})^2 : (b^{2m-1})^3 = b^{6m} : b^{3(2m-1)} = b^{6m} : b^{6m-3} = b^{6m-(6m-3)} = b^{6m-6m+3} = b^3 \]

4. \[ \left(-3\frac{1}{3} \alpha^2 b^6\right)^2 = \left(-\frac{10}{3} \alpha^2 b^6\right)^2 = \left(-\frac{10}{3}\right)^2 (\alpha^2)^2 (b^6)^2 = \frac{100}{9} \alpha^4 b^{12} \]

5. \[ \left(-\frac{1}{2} n^5 m^3\right)^4 = \left(-\frac{1}{2}\right)^4 (n^5)^4 (m^3)^4 = \frac{1}{16} n^{20} m^{12} \]

6. \[ \frac{5^6 \cdot 25^2}{125^3} = \frac{5^6 \cdot (5^2)^2}{(5^3)^3} = \frac{5^6 \cdot 5^4}{5^9} = \frac{5^{6+4}}{5^9} = \frac{5^{10}}{5^9} = 5^{10-9} = 5 \]

7. \[ \frac{27^4 \cdot 3^2}{81^3} = \frac{(3^3)^4 \cdot 3^2}{(3^4)^3} = \frac{3^{12} \cdot 3^2}{3^{12}} = \frac{3^{14}}{3^{12}} = 3^{14-12} = 3^2 = 9 \]

8. \[ \frac{16^3 \cdot 8^2}{64^3} = \frac{(2^4)^3 \cdot (2^3)^2}{(2^6)^3} = \frac{2^{12} \cdot 2^6}{2^{18}} = \frac{2^{18}}{2^{18}} = 1 \]

9. \[ \frac{64^2 \cdot 4^3}{16^4} = \frac{(4^3)^2 \cdot 4^3}{(4^2)^4} = \frac{4^6 \cdot 4^3}{4^8} = \frac{4^9}{4^8} = 4^{9-8} = 4 \]

10. \[ \frac{2^8 \cdot (2^3)^2}{2^{10}} = \frac{2^8 \cdot 2^6}{2^{10}} = \frac{2^{14}}{2^{10}} = 2^{14-10} = 2^4 = 16 \]

11. \[ \frac{6^{12}}{(6^5)^2 \cdot 6^0} = \frac{6^{12}}{6^{10} \cdot 1} = \frac{6^{12}}{6^{10}} = 6^{12-10} = 6^2 = 36 \]

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. \[ (a^{m+1})^2 : a^{m-1} = a^{2(m+1)} : a^{m-1} = a^{2m+2} : a^{m-1} = a^{(2m+2)-(m-1)} = a^{2m+2-m+1} = a^{m+3} \]

15. \[ a^{5n+3} : (a^n)^4 = a^{5n+3} : a^{4n} = a^{5n+3-4n} = a^{n+3} \]