Решение задач на действия со степенями

Ниже представлены решения заданий из таблицы. Для удобства переписывания в тетрадь решения сгруппированы по столбцам. Столбец A 1. \(x^8 \cdot x^2 = x^{8+2} = x^{10}\) 2. \(x^8 : x^2 = x^{8-2} = x^6\) 3. \((x^8)^2 = x^{8 \cdot 2} = x^{16}\) 4. \(x^8 + x^2\) — упростить нельзя (разные степени). 5. \(x^8 - x^2\) — упростить нельзя. 6. \(x^8 + x^8 = 2x^8\) 7. \(\frac{x^2 \cdot x^8}{x} = \frac{x^{10}}{x^1} = x^9\) 8. \(\frac{(x^2)^3 \cdot x^5}{x^{10}} = \frac{x^6 \cdot x^5}{x^{10}} = \frac{x^{11}}{x^{10}} = x\) 9. \(3^8 \cdot (\frac{1}{3})^8 = (3 \cdot \frac{1}{3})^8 = 1^8 = 1\) 10. \(2^5 \cdot 4^6 = 2^5 \cdot (2^2)^6 = 2^5 \cdot 2^{12} = 2^{17}\) 11. \(49^4 : 7^2 = (7^2)^4 : 7^2 = 7^8 : 7^2 = 7^6\) 12. \(25^4 : 5^4 = (25 : 5)^4 = 5^4 = 625\) 13. \(\frac{3^8}{3^9} = 3^{8-9} = 3^{-1} = \frac{1}{3}\) 14. \(\frac{25^4}{5^{10}} = \frac{(5^2)^4}{5^{10}} = \frac{5^8}{5^{10}} = 5^{-2} = \frac{1}{25}\) 15. \(8^2 \cdot 5^4 = (2^3)^2 \cdot 5^4 = 2^6 \cdot 5^4 = 64 \cdot 625 = 40000\) 16. \(4^n \cdot 4^{n+1} = 4^{n+n+1} = 4^{2n+1}\) 17. \(4^{3+n} \cdot 4^3 = 4^{3+n+3} = 4^{n+6}\) 18. \((4^n)^5 : 4^n = 4^{5n} : 4^n = 4^{4n}\) 19. \(4^n + 4^n = 2 \cdot 4^n = 2 \cdot (2^2)^n = 2 \cdot 2^{2n} = 2^{2n+1}\) 20. \(4^n \cdot 4^n = 4^{2n}\) Столбец B 1. \(a^5 \cdot a^6 = a^{11}\) 2. \((a^5)^6 = a^{30}\) 3. \(a^7 : a = a^6\) 4. \(a^{50} : a^{10} = a^{40}\) 5. \((a^2)^4 \cdot a^3 = a^8 \cdot a^3 = a^{11}\) 6. \(3^9 : (3^2)^4 = 3^9 : 3^8 = 3^1 = 3\) 7. \(6^7 \cdot 6^2 \cdot 6 = 6^{7+2+1} = 6^{10}\) 8. \(\frac{3^5}{3^4} = 3^1 = 3\) 9. \(\frac{4^8}{4^{10}} = 4^{-2} = \frac{1}{16}\) 10. \(\frac{(7^2)^3 \cdot 7}{7^4 \cdot 7^2} = \frac{7^6 \cdot 7^1}{7^6} = 7^1 = 7\) 11. \(\frac{5^4 \cdot (5^2)^4}{5^6 \cdot (5^1)^3} = \frac{5^4 \cdot 5^8}{5^6 \cdot 5^3} = \frac{5^{12}}{5^9} = 5^3 = 125\) 12. \(2^6 - 2^3 = 64 - 8 = 56\) 13. \(2^9 : 2^3 = 2^6 = 64\) 14. \(6^4 \cdot 36 = 6^4 \cdot 6^2 = 6^6\) 15. \(5^7 \cdot 125 = 5^7 \cdot 5^3 = 5^{10}\) 16. \(32 \cdot 2^{15} = 2^5 \cdot 2^{15} = 2^{20}\) 17. \(3^n \cdot 3^{n+4} = 3^{2n+4}\) 18. \(5^{6n} : 5^n = 5^{5n}\) 19. \(3^{2n} \cdot 27 = 3^{2n} \cdot 3^3 = 3^{2n+3}\) 20. \(6^{n+8} \cdot 6^{2n-8} = 6^{n+8+2n-8} = 6^{3n}\) Столбец C 1. \((a^4)^2 = a^8\) 2. \(a^4 \cdot a^4 = a^8\) 3. \(a^4 : a^4 = 1\) 4. \((a^4)^4 = a^{16}\) 5. \(a^4 + a^4 = 2a^4\) 6. \(a^7 + a^3\) — упростить нельзя. 7. \(a \cdot a \cdot a = a^3\) 8. \(\frac{5^7}{5^5} = 5^2 = 25\) 9. \(\frac{(5^2)^3 \cdot 5}{5^4} = \frac{5^6 \cdot 5}{5^4} = \frac{5^7}{5^4} = 5^3 = 125\) 10. \((2c)^5 : c^5 = 2^5 \cdot c^5 : c^5 = 2^5 = 32\) 11. \(100 \cdot 10^7 = 10^2 \cdot 10^7 = 10^9\) 12. \(\frac{625 \cdot 5^5}{5^{10}} = \frac{5^4 \cdot 5^5}{5^{10}} = \frac{5^9}{5^{10}} = 5^{-1} = 0,2\) 13. \(\frac{27^2 \cdot 3^4}{9^3} = \frac{(3^3)^2 \cdot 3^4}{(3^2)^3} = \frac{3^6 \cdot 3^4}{3^6} = 3^4 = 81\) 14. \(10^4 \cdot 10^2 \cdot 10 = 10^7\) 15. \(9^7 \cdot 2^7 = (9 \cdot 2)^7 = 18^7\) 16. \(c^8 \cdot c^{n-8} = c^{8+n-8} = c^n\) 17. \(c^{2n} \cdot c^n = c^{3n}\) 18. \(c^n \cdot c^n = c^{2n}\) 19. \(c^n + c^n = 2c^n\) 20. \((c^n)^2 : c^3 = c^{2n-3}\) Столбец D 1. \(c^9 : c^3 = c^6\) 2. \(c^9 - c^3\) — упростить нельзя. 3. \((c^9)^3 = c^{27}\) 4. \(c^9 + c^3\) — упростить нельзя. 5. \(\frac{c^9}{c^3} = c^6\) 6. \((c^5)^4 \cdot c^4 = c^{20} \cdot c^4 = c^{24}\) 7. \((c^8)^2 : (c^4)^4 = c^{16} : c^{16} = 1\) 8. \(\frac{c^5 \cdot c^8}{c^4 \cdot c^{10}} = \frac{c^{13}}{c^{14}} = c^{-1} = \frac{1}{c}\) 9. \(\frac{(c^3)^4 \cdot c}{c^3 \cdot (c^2)^2} = \frac{c^{12} \cdot c}{c^3 \cdot c^4} = \frac{c^{13}}{c^7} = c^6\) 10. \((3c)^4 : c^4 = 3^4 \cdot c^4 : c^4 = 3^4 = 81\) 11. \(3c^4 : c^4 = 3\) 12. \(c^0 \cdot c^m = 1 \cdot c^m = c^m\) 13. \(c^2 \cdot c^2 \cdot c^2 = c^6\) 14. \(27 \cdot 3^{10} = 3^3 \cdot 3^{10} = 3^{13}\) 15. \(125^4 : 5^6 = (5^3)^4 : 5^6 = 5^{12} : 5^6 = 5^6 = 15625\) 16. \(\frac{8^{12} \cdot 4}{2^{10}} = \frac{(2^3)^{12} \cdot 2^2}{2^{10}} = \frac{2^{36} \cdot 2^2}{2^{10}} = \frac{2^{38}}{2^{10}} = 2^{28}\) 17. \(y^n \cdot y^{5n} = y^{6n}\) 18. \(y^n \cdot y^2 = y^{n+2}\) 19. \((y^{n+2})^3 : y = y^{3n+6} : y^1 = y^{3n+5}\) 20. \(y^n \cdot y^{4-n} = y^{n+4-n} = y^4\)