Вариант 2 1. Преобразуйте выражение: 1) x12x3 2) a-3. a8 3) t-5-t 4) y4y-4 5) m³ : m5 6) n-12: n-8 7) (s3)7 8) (x-4)5 9) (k-7)-2 -3 10) (xy)-411) () 2. Вычислите: 1) 24: 28 2) (4-1)-2 3) 14-7:14-8 4) (a-5)4.a22 3. Упростите: 1) 140 18x-6 7y-5 2) 18.72 y5 6x-12 4. Вычислить: b-2 b-3 56a4 1) (7-11)2 7-23 2-7.217 5) 28 -9 2) 3-11. (35)26) (62) 6-20 3) 8-6.8-7 8-12 7) 7-6(72)4 4) (7.103)2 (16·10-4) 612.1110 8) 6610

1. Преобразуйте выражение:
   1. \[x^{12} \cdot x^3 = x^{12+3} = x^{15}\]
   2. \[a^{-3} \cdot a^8 = a^{-3+8} = a^5\]
   3. \[t^{-5} \cdot t = t^{-5+1} = t^{-4}\]
   4. \[y^4 \cdot y^{-4} = y^{4+(-4)} = y^0 = 1\]
   5. \[m^9 : m^5 = m^{9-5} = m^4\]
   6. \[n^{-12} : n^{-8} = n^{-12 - (-8)} = n^{-12+8} = n^{-4}\]
   7. \[(s^3)^7 = s^{3 \cdot 7} = s^{21}\]
   8. \[(x^{-4})^5 = x^{-4 \cdot 5} = x^{-20}\]
   9. \[(k^{-7})^{-2} = k^{-7 \cdot (-2)} = k^{14}\]
   10. \[(x^6y)^{-4} = (x^6)^{-4} \cdot y^{-4} = x^{6 \cdot (-4)} \cdot y^{-4} = x^{-24}y^{-4}\]
   11. \[\left(\frac{a^{-4}}{b}\right)^{-3} = \frac{(a^{-4})^{-3}}{b^{-3}} = \frac{a^{-4 \cdot (-3)}}{b^{-3}} = \frac{a^{12}}{b^{-3}} = a^{12}b^3\]
2. Вычислите:
   1. \[\frac{2^4}{2^8} = 2^{4-8} = 2^{-4} = \frac{1}{2^4} = \frac{1}{16}\]
   2. \[(4^{-1})^{-2} = 4^{-1 \cdot (-2)} = 4^2 = 16\]
   3. \[\frac{14^{-7}}{14^{-8}} = 14^{-7 - (-8)} = 14^{-7+8} = 14^1 = 14\]
   4. \[(a^{-5})^4 \cdot a^{22} = a^{-5 \cdot 4} \cdot a^{22} = a^{-20} \cdot a^{22} = a^{-20+22} = a^2\]
3. Упростите:
   1. \[\frac{14a}{b^{-3}} \cdot \frac{b^{-2}}{56a^4} = \frac{14}{56} \cdot \frac{a}{a^4} \cdot \frac{b^{-2}}{b^{-3}} = \frac{1}{4} \cdot a^{1-4} \cdot b^{-2 - (-3)} = \frac{1}{4} \cdot a^{-3} \cdot b^{-2+3} = \frac{1}{4} \cdot a^{-3} \cdot b^1 = \frac{b}{4a^3}\]
   2. \[\frac{18x^{-6}}{y^5} \cdot \frac{7y^{-5}}{6x^{-12}} = \frac{18 \cdot 7}{6} \cdot \frac{x^{-6}}{x^{-12}} \cdot \frac{y^{-5}}{y^5} = 3 \cdot 7 \cdot x^{-6 - (-12)} \cdot y^{-5-5} = 21 \cdot x^{-6+12} \cdot y^{-10} = 21x^6y^{-10} = \frac{21x^6}{y^{10}}\]
4. Вычислите:
   1. \[\frac{(7^{-11})^2}{7^{-23}} = \frac{7^{-11 \cdot 2}}{7^{-23}} = \frac{7^{-22}}{7^{-23}} = 7^{-22 - (-23)} = 7^{-22+23} = 7^1 = 7\]
   2. \[3^{-11} \cdot (3^5)^2 = 3^{-11} \cdot 3^{5 \cdot 2} = 3^{-11} \cdot 3^{10} = 3^{-11+10} = 3^{-1} = \frac{1}{3}\]
   3. \[\frac{8^{-6} \cdot 8^{-7}}{8^{-12}} = \frac{8^{-6+(-7)}}{8^{-12}} = \frac{8^{-13}}{8^{-12}} = 8^{-13 - (-12)} = 8^{-13+12} = 8^{-1} = \frac{1}{8}\]
   4. \[(7 \cdot 10^3)^2 \cdot (16 \cdot 10^{-4}) = 7^2 \cdot (10^3)^2 \cdot 16 \cdot 10^{-4} = 49 \cdot 10^{3 \cdot 2} \cdot 16 \cdot 10^{-4} = 49 \cdot 10^6 \cdot 16 \cdot 10^{-4} = 49 \cdot 16 \cdot 10^{6+(-4)} = 784 \cdot 10^2 = 78400\]
   5. \[\frac{2^{-7} \cdot 2^{17}}{2^8} = \frac{2^{-7+17}}{2^8} = \frac{2^{10}}{2^8} = 2^{10-8} = 2^2 = 4\]
   6. \[\frac{(6^2)^{-9}}{6^{-20}} = \frac{6^{2 \cdot (-9)}}{6^{-20}} = \frac{6^{-18}}{6^{-20}} = 6^{-18 - (-20)} = 6^{-18+20} = 6^2 = 36\]
   7. \[7^{-6} \cdot (7^2)^4 = 7^{-6} \cdot 7^{2 \cdot 4} = 7^{-6} \cdot 7^8 = 7^{-6+8} = 7^2 = 49\]
   8. \[\frac{6^{12} \cdot 11^{10}}{66^{10}} = \frac{6^{12} \cdot 11^{10}}{(6 \cdot 11)^{10}} = \frac{6^{12} \cdot 11^{10}}{6^{10} \cdot 11^{10}} = \frac{6^{12}}{6^{10}} \cdot \frac{11^{10}}{11^{10}} = 6^{12-10} \cdot 1 = 6^2 = 36\]

Ответ: Решения выше.