10. Возведение в степень произведения и степени. A (ab) (xy)7 (2y)5 (1/3ab)3 (3 * 10)4 (-2x3y2)5 (b4)3 (xyz2)6 (2x3)4 (m5)n ((x2)4)7 (p3)5 : p10 (y2)7 * y3 (a4)5 : (a3)6 b13 : (b2)6 B (3x2y4)3 (-2x3y5)6 b19: (b3b2)3 (p2/4a)3 (a2 * b3/2p)2 (-1/3a4b5)3 (-2 1/3x5y3)2 (2 1/2n6m4)4 (7^9 * 7^2 / 7^10)^2 (-2x6y4)3 a18 : (a3)5 * a0 (b10 * b2)3 : b20 (a6)2 : (a2)4 * a5 ((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 (-3 1/3a2b6)2 (-1 1/2n5m3)4 5^6 * 25^2 / 125^3 27^4 * 3^2 / 81^3 16^3 * 8^2 / 64^3 64^2 * 4^3 / 16^4 2^8 * (2^3)2 : 2^10 6^12 : (6^5)2 * 6^0 (c4n+1)3 : (c6n-2)2 (xn-3 * xn+2)2 (am+1)2: am-1 a5n+3: (an)4

Давай разберем это задание по алгебре. Здесь нужно упростить выражения, используя свойства степеней. Начнем с первого столбца (A), затем перейдем к (B) и (C).

Столбец A

1. (ab)⁷ = a⁷b⁷

2. (xy)⁷ = x⁷y⁷

3. (2y)⁵ = 2⁵y⁵ = 32y⁵

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 = 3⁴ \cdot 10⁴ = 81 \cdot 10000 = 810000

6. (-2x³y²)⁵ = (-2)⁵x^3*5y^2*5 = -32x¹⁵y¹⁰

7. (b⁴)³ = b^4*3 = b¹²

8. (xyz²)⁶ = x⁶y⁶z^2*6 = x⁶y⁶z¹²

9. (2x³)⁴ = 2⁴x^3*4 = 16x¹²

10. (m⁵)ⁿ = m⁵ⁿ

11. ((x²)⁴)⁷ = (x^2*4)⁷ = x^8*7 = x⁵⁶

12. (p³)⁵ : p¹⁰ = p^3*5 : p¹⁰ = p¹⁵ : p¹⁰ = p^15-10 = p⁵

13. (y²)⁷ \cdot y³ = y^2*7 \cdot y³ = y¹⁴ \cdot y³ = y¹⁴⁺³ = y¹⁷

14. (a⁴)⁵ : (a³)⁶ = a^4*5 : a^3*6 = a²⁰ : a¹⁸ = a^20-18 = a²

15. b¹³ : (b²)⁶ = b¹³ : b^2*6 = b¹³ : b¹² = b^13-12 = b

Столбец B

1. (3x²y⁴)³ = 3³x^2*3y^4*3 = 27x⁶y¹²

2. (-2x³y⁵)⁶ = (-2)⁶x^3*6y^5*6 = 64x¹⁸y³⁰

3. b¹⁹ : (b³b²)³ = b¹⁹ : (b³⁺²)³ = b¹⁹ : (b⁵)³ = b¹⁹ : b^5*3 = b¹⁹ : b¹⁵ = b^19-15 = b⁴

4. \(\left(\frac{p^2}{4a}\right)^3 = \frac{(p^2)^3}{(4a)^3} = \frac{p^{2*3}}{4^3a^3} = \frac{p^6}{64a^3}\)

5. \(\left(\frac{a^2 \cdot b^3}{2p}\right)^2 = \frac{(a^2 \cdot b^3)^2}{(2p)^2} = \frac{(a^2)^2 \cdot (b^3)^2}{2^2p^2} = \frac{a^{2*2} \cdot b^{3*2}}{4p^2} = \frac{a^4b^6}{4p^2}\)

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

7. \(\left(-2\frac{1}{3}x^5y^3\right)^2 = \left(-\frac{7}{3}x^5y^3\right)^2 = \left(-\frac{7}{3}\right)^2(x^5)^2(y^3)^2 = \frac{49}{9}x^{10}y^6\)

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

10. (-2x⁶y⁴)³ = (-2)³(x⁶)³(y⁴)³ = -8x¹⁸y¹²

11. a¹⁸ : (a³)⁵ \cdot a⁰ = a¹⁸ : a¹⁵ \cdot 1 = a^18-15 \cdot 1 = a³

12. (b¹⁰ \cdot b²)³ : b²⁰ = (b¹⁰⁺²)³ : b²⁰ = (b¹²)³ : b²⁰ = b³⁶ : b²⁰ = b^36-20 = b¹⁶

13. (a⁶)² : (a²)⁴ \cdot a⁵ = a¹² : a⁸ \cdot a⁵ = a^12-8 \cdot a⁵ = a⁴ \cdot a⁵ = a⁴⁺⁵ = a⁹

14. ((p²)³)⁵ = (p⁶)⁵ = p³⁰

15. (x³)⁸ : (x⁴)⁶ = x²⁴ : x²⁴ = 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 = a⁴

2. (cⁿ⁺¹)⁴ : (c^n-2)³ = c⁴⁽ⁿ⁺¹⁾ : c^3(n-2) = c⁴ⁿ⁺⁴ : c^3n-6 = c^4n+4-(3n-6) = c^4n+4-3n+6 = cⁿ⁺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 = b³

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

5. \(\left(-1\frac{1}{2}n^5m^3\right)^4 = \left(-\frac{3}{2}n^5m^3\right)^4 = \left(-\frac{3}{2}\right)^4(n^5)^4(m^3)^4 = \frac{81}{16}n^{20}m^{12}\)

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

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. \(2^8 \cdot (2^3)^2 : 2^{10} = 2^8 \cdot 2^6 : 2^{10} = 2^{8+6} : 2^{10} = 2^{14} : 2^{10} = 2^{14-10} = 2^4 = 16\)

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

12. (c⁴ⁿ⁺¹)³ : (c^6n-2)² = c^3(4n+1) : c^2(6n-2) = c¹²ⁿ⁺³ : c^12n-4 = c^{12n+3-(12n-4)} = c^12n+3-12n+4 = c⁷

13. (x^n-3 \cdot xⁿ⁺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. a⁵ⁿ⁺³ : (aⁿ)⁴ = a⁵ⁿ⁺³ : a⁴ⁿ = a^5n+3-4n = aⁿ⁺³

Ответ: Выше приведены упрощенные выражения для каждого элемента.