ВОЗВЕДЕНИЕ В СТЕПЕНЬ ПРОИЗВЕДЕНИЯ И СТЕПЕНИ Важно знать: (a · b)n = an · bn (am)n = amn. Выполните возведение в степень 1) (62)3 = 62·3 = 66 2) (a5)10 = a5·10 = a50 3) (c4)12 = c4·12 = c48 4) (k5)5 = k5·5 = k25 5) (p4)5 = p4·5 = p20 6) (y10)2 = y10·2 = y20 7) (m3)6 = m3·6 = m18 8) (x15)3 = x15·3 = x45 9) (q7)6 = q7·6 = q42 10) (n25)4 = n25·4 = n100 11) (t5)20 = t5·20 = t100 12) (c4)16 = c4·16 = c64 13) (k13)3 = k13·3 = k39 14) (a11)5 = a11·5 = a55 15) (k3n)2 = k2n2 Представьте степень в виде произведения степеней 16) (ab)5 = a5 · b5 17) (xy)3 = x3 · y3 18) (an)2 = a2 · n2 19) (2a)5 = 25 · a5 20) (5c)2 = 52 · c2 21) (3ab)4 = 34 · a4 · b4 22) (10xy)5 = 105 · x5 · y5 23) (4mn)3 = 43 · m3 · n3 24) (abc)7 = a7 · b7 · c7 25) (xyz)3 = x3 · y3 · z3 26) (2x2)3 = 23 · (x2)3 27) (4a3)2 = 42 · (a3)2 28) (9m2)2 = 92 · (m2)2 29) (-5n3)2 = (-5)2 · (n3)2 30) (-2a6)3 = (-2)3 · (a6)3 Найдите значение выражения 31) 25 · 55 = (2 · 5)5 = 105 = 100000 32) 146 · (1/7)6 33) 0,259 · 49 34) (22)3 : 23 = 26 : 23 = 23 = 8 35) (210)2 : 27 / 27 = 220 : 27 / 27 = 220-7-7 = 26 36) 513 · 57 / (54)5 = 513+7 / 520 = 520 / 520 = 1 37) 310 · (33)3 / 316 = 310 · 39 / 316 = 319 / 316 = 33 = 27 38) (24)3 · (25)2 / (23)6 = 212 · 210 / 218 = 222 / 218 = 24 = 16 39) 23 · 43 = 23 · (22)3 = 23 · 26 = 29 = 512 40) 162 · 43 : 84 = 24 · 26 : 212 = 210 : 212 = 2-2 = 1/4 41) 625 · 57 : 1253 = 54 · 57 : (53)3 = 54+7 : 59 = 511 : 59 = 52 = 25 42) 1003 · 1012 : 10004 = (102)3 · 1012 : (103)4 = 106 · 1012 : 1012 = 106 43) (22)9 · 8 / 220 = 218 · 23 / 220 = 221 / 220 = 21 = 2 44) 27 · (34)6 / (320 · 35) = 33 · 324 / (320 · 35) = 327 / 325 = 32 = 9 45) 515 · 512 / (125 · (58)3) = 515+12 / (53 · 524) = 527 / 527 = 1

Задание 36

Важно знать:

\( (a \cdot b)^n = a^n \cdot b^n \)

\( (a^m)^n = a^{m \cdot n} \)

Выполните возведение в степень

1. \( (6^2)^3 = 6^{2 \cdot 3} = 6^6 \)
2. \( (a^5)^{10} = a^{5 \cdot 10} = a^{50} \)
3. \( (c^4)^{12} = c^{4 \cdot 12} = c^{48} \)
4. \( (k^5)^5 = k^{5 \cdot 5} = k^{25} \)
5. \( (p^4)^5 = p^{4 \cdot 5} = p^{20} \)
6. \( (y^{10})^2 = y^{10 \cdot 2} = y^{20} \)
7. \( (m^3)^6 = m^{3 \cdot 6} = m^{18} \)
8. \( (x^{15})^3 = x^{15 \cdot 3} = x^{45} \)
9. \( (q^7)^6 = q^{7 \cdot 6} = q^{42} \)
10. \( (n^{25})^4 = n^{25 \cdot 4} = n^{100} \)
11. \( (t^5)^{20} = t^{5 \cdot 20} = t^{100} \)
12. \( (c^4)^{16} = c^{4 \cdot 16} = c^{64} \)
13. \( (k^{13})^3 = k^{13 \cdot 3} = k^{39} \)
14. \( (a^{11})^5 = a^{11 \cdot 5} = a^{55} \)
15. \( (k^3 n)^2 = (k^3)^2 n^2 = k^6 n^2 \)

Представьте степень в виде произведения степеней

16. \( (ab)^5 = a^5 \cdot b^5 \)
17. \( (xy)^3 = x^3 \cdot y^3 \)
18. \( (an)^2 = a^2 \cdot n^2 \)
19. \( (2a)^5 = 2^5 \cdot a^5 \)
20. \( (5c)^2 = 5^2 \cdot c^2 \)
21. \( (3ab)^4 = 3^4 \cdot a^4 \cdot b^4 \)
22. \( (10xy)^5 = 10^5 \cdot x^5 \cdot y^5 \)
23. \( (4mn)^3 = 4^3 \cdot m^3 \cdot n^3 \)
24. \( (abc)^7 = a^7 \cdot b^7 \cdot c^7 \)
25. \( (xyz)^3 = x^3 \cdot y^3 \cdot z^3 \)
26. \( (2x^2)^3 = 2^3 \cdot (x^2)^3 = 8x^6 \)
27. \( (4a^3)^2 = 4^2 \cdot (a^3)^2 = 16a^6 \)
28. \( (9m^2)^2 = 9^2 \cdot (m^2)^2 = 81m^4 \)
29. \( (-5n^3)^2 = (-5)^2 \cdot (n^3)^2 = 25n^6 \)
30. \( (-2a^6)^3 = (-2)^3 \cdot (a^6)^3 = -8a^{18} \)

Найдите значение выражения

31. \( 2^5 \cdot 5^5 = (2 \cdot 5)^5 = 10^5 = 100000 \)
32. \( 14^6 \cdot (\frac{1}{7})^6 = (14 \cdot \frac{1}{7})^6 = 2^6 = 64 \)
33. \( 0.25^9 \cdot 4^9 = (0.25 \cdot 4)^9 = 1^9 = 1 \)
34. \( (2^2)^3 : 2^3 = 2^{2 \cdot 3} : 2^3 = 2^6 : 2^3 = 2^{6-3} = 2^3 = 8 \)
35. \( \frac{(2^{10})^2 : 2^7}{2^7} = \frac{2^{10 \cdot 2} : 2^7}{2^7} = \frac{2^{20} : 2^7}{2^7} = \frac{2^{20-7}}{2^7} = \frac{2^{13}}{2^7} = 2^{13-7} = 2^6 = 64 \)
36. \( \frac{5^{13} \cdot 5^7}{(5^4)^5} = \frac{5^{13+7}}{5^{4 \cdot 5}} = \frac{5^{20}}{5^{20}} = 5^{20-20} = 5^0 = 1 \)
37. \( \frac{3^{10} \cdot (3^3)^3}{3^{16}} = \frac{3^{10} \cdot 3^{3 \cdot 3}}{3^{16}} = \frac{3^{10} \cdot 3^9}{3^{16}} = \frac{3^{10+9}}{3^{16}} = \frac{3^{19}}{3^{16}} = 3^{19-16} = 3^3 = 27 \)
38. \( \frac{(2^4)^3 \cdot (2^5)^2}{(2^3)^6} = \frac{2^{4 \cdot 3} \cdot 2^{5 \cdot 2}}{2^{3 \cdot 6}} = \frac{2^{12} \cdot 2^{10}}{2^{18}} = \frac{2^{12+10}}{2^{18}} = \frac{2^{22}}{2^{18}} = 2^{22-18} = 2^4 = 16 \)
39. \( 2^3 \cdot 4^3 = 2^3 \cdot (2^2)^3 = 2^3 \cdot 2^{2 \cdot 3} = 2^3 \cdot 2^6 = 2^{3+6} = 2^9 = 512 \)
40. \( 16^2 \cdot 4^3 : 8^4 = (2^4)^2 \cdot (2^2)^3 : (2^3)^4 = 2^{4 \cdot 2} \cdot 2^{2 \cdot 3} : 2^{3 \cdot 4} = 2^8 \cdot 2^6 : 2^{12} = 2^{8+6} : 2^{12} = 2^{14} : 2^{12} = 2^{14-12} = 2^2 = 4 \)
41. \( 625 \cdot 5^7 : 125^3 = 5^4 \cdot 5^7 : (5^3)^3 = 5^{4+7} : 5^{3 \cdot 3} = 5^{11} : 5^9 = 5^{11-9} = 5^2 = 25 \)
42. \( 100^3 \cdot 10^{12} : 1000^4 = (10^2)^3 \cdot 10^{12} : (10^3)^4 = 10^{2 \cdot 3} \cdot 10^{12} : 10^{3 \cdot 4} = 10^6 \cdot 10^{12} : 10^{12} = 10^{6+12} : 10^{12} = 10^{18} : 10^{12} = 10^{18-12} = 10^6 \)
43. \( \frac{(2^2)^9 \cdot 8}{2^{20}} = \frac{2^{2 \cdot 9} \cdot 2^3}{2^{20}} = \frac{2^{18} \cdot 2^3}{2^{20}} = \frac{2^{18+3}}{2^{20}} = \frac{2^{21}}{2^{20}} = 2^{21-20} = 2^1 = 2 \)
44. \( \frac{27 \cdot (3^4)^6}{3^{20} \cdot 3^5} = \frac{3^3 \cdot 3^{4 \cdot 6}}{3^{20} \cdot 3^5} = \frac{3^3 \cdot 3^{24}}{3^{20+5}} = \frac{3^{3+24}}{3^{25}} = \frac{3^{27}}{3^{25}} = 3^{27-25} = 3^2 = 9 \)
45. \( \frac{5^{15} \cdot 5^{12}}{125 \cdot (5^8)^3} = \frac{5^{15+12}}{5^3 \cdot 5^{8 \cdot 3}} = \frac{5^{27}}{5^3 \cdot 5^{24}} = \frac{5^{27}}{5^{3+24}} = \frac{5^{27}}{5^{27}} = 5^{27-27} = 5^0 = 1 \)