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Ответ:
2. Вычислите:
- a) \( 3^7 \cdot 3^6 = 3^{7+6} = 3^{13} \)
б) \( 7^9 \cdot 7^8 = 7^{9+8} = 7^{17} \)
в) \( \left(\frac{1}{5}\right)^{-7} \cdot \left(\frac{1}{5}\right)^6 = \left(\frac{1}{5}\right)^{-7+6} = \left(\frac{1}{5}\right)^{-1} = 5 \)
г) \( \left(\frac{1}{4}\right)^{14} \cdot \left(\frac{1}{4}\right)^{-16} = \left(\frac{1}{4}\right)^{14-16} = \left(\frac{1}{4}\right)^{-2} = 4^2 = 16 \) - a) \( 2^5 : 2^6 = 2^{5-6} = 2^{-1} = \frac{1}{2} \)
б) \( 5^5 : 5^{-2} = 5^{5-(-2)} = 5^{5+2} = 5^7 \)
в) \( 6^{-9} : 6^{-11} = 6^{-9-(-11)} = 6^{-9+11} = 6^2 = 36 \)
г) \( \left(\frac{1}{7}\right)^{-3} : \left(\frac{1}{7}\right)^{-3} = \left(\frac{1}{7}\right)^{-3-(-3)} = \left(\frac{1}{7}\right)^0 = 1 \) - a) \( (3^2)^{-1} = 3^{2 \cdot (-1)} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \)
б) \( \left(\left(\frac{1}{2}\right)^3\right)^{-2} = \left(\frac{1}{2}\right)^{3 \cdot (-2)} = \left(\frac{1}{2}\right)^{-6} = 2^6 = 64 \)
в) \( (0.1^{-2})^6 = 0.1^{-2 \cdot 6} = 0.1^{-12} = 10^{12} \)
г) \( \left(\frac{1}{6}\right)^0 = 1 \) - a) \( -17 \cdot 34^{-1} = -17 \cdot \frac{1}{34} = -\frac{17}{34} = -\frac{1}{2} \)
б) \( -10 \cdot 2^{-3} = -10 \cdot \frac{1}{2^3} = -10 \cdot \frac{1}{8} = -\frac{10}{8} = -\frac{5}{4} \)
в) \( \left(\frac{1}{8}\right)^{-2} - 0.01^{-1} = 8^2 - \frac{1}{0.01} = 64 - 100 = -36 \)
г) \( 6^{-2} + 24^{-1} = \frac{1}{6^2} + \frac{1}{24} = \frac{1}{36} + \frac{1}{24} = \frac{2}{72} + \frac{3}{72} = \frac{5}{72} \) - a) \( 32 \cdot 2^{-6} = 2^5 \cdot 2^{-6} = 2^{5-6} = 2^{-1} = \frac{1}{2} \)
б) \( 27 \cdot (3^{-2})^2 = 3^3 \cdot 3^{-4} = 3^{3-4} = 3^{-1} = \frac{1}{3} \)
в) \( 7^{-8} \cdot 7^9 : 49 = 7^{-8+9} : 7^2 = 7^1 : 7^2 = 7^{1-2} = 7^{-1} = \frac{1}{7} \)
г) \( 25^{-2} \cdot \left(\frac{1}{5}\right)^{-6} = (5^2)^{-2} \cdot 5^6 = 5^{-4} \cdot 5^6 = 5^{-4+6} = 5^2 = 25 \) - a) \( 81^{-2} \cdot 27^2 = (3^4)^{-2} \cdot (3^3)^2 = 3^{-8} \cdot 3^6 = 3^{-8+6} = 3^{-2} = \frac{1}{9} \)
б) \( 16^{-5} : 8^{-6} = (2^4)^{-5} : (2^3)^{-6} = 2^{-20} : 2^{-18} = 2^{-20 - (-18)} = 2^{-20+18} = 2^{-2} = \frac{1}{4} \)
в) \( \frac{(-6)^{-9} \cdot 6^{-7}}{6^{-15}} = \frac{-1 \cdot 6^{-9} \cdot 6^{-7}}{6^{-15}} = - \frac{6^{-9-7}}{6^{-15}} = - \frac{6^{-16}}{6^{-15}} = -6^{-16 - (-15)} = -6^{-16+15} = -6^{-1} = -\frac{1}{6} \)
г) \( \frac{4^{-6} \cdot 16^{-5}}{8^{-10}} = \frac{(2^2)^{-6} \cdot (2^4)^{-5}}{(2^3)^{-10}} = \frac{2^{-12} \cdot 2^{-20}}{2^{-30}} = \frac{2^{-12-20}}{2^{-30}} = \frac{2^{-32}}{2^{-30}} = 2^{-32 - (-30)} = 2^{-32+30} = 2^{-2} = \frac{1}{4} \)
3. Упростите выражение:
- a) \( 6x^{-5}y^7 \cdot 2.5x^7y^{-6} = (6 \cdot 2.5) x^{-5+7} y^{7-6} = 15 x^2 y^1 = 15x^2y \)
б) \( 0.8a^{-6}b^4 \cdot 5a^{12}b^{-4} = (0.8 \cdot 5) a^{-6+12} b^{4-4} = 4 a^6 b^0 = 4a^6 \) - a) \( 3.2a^6b : (0.8a^3b^{-3}) = \frac{3.2}{0.8} \frac{a^6}{a^3} \frac{b}{b^{-3}} = 4 a^{6-3} b^{1-(-3)} = 4 a^3 b^4 \)
б) \( 3\frac{1}{2}m^{-8}n^7 : \left(-\frac{7}{8}m^{-5}n^{-7}\right) = \frac{7}{2}m^{-8}n^7 \cdot \left(-\frac{8}{7}m^5n^7\right) = \left(\frac{7}{2} \cdot -\frac{8}{7}\right) m^{-8+5} n^{7+7} = -4 m^{-3} n^{14} = -\frac{4n^{14}}{m^3} \) - a) \( \frac{13x^{-4} \cdot y}{y^{-6} \cdot 52x^{-5}} = \frac{13}{52} \frac{x^{-4}}{x^{-5}} \frac{y}{y^{-6}} = \frac{1}{4} x^{-4-(-5)} y^{1-(-6)} = \frac{1}{4} x^1 y^7 = \frac{1}{4}xy^7 \)
б) \( \frac{21a^{-4} \cdot 5b^{-6}}{10b^6 \cdot 7a^{-8}} = \frac{21 \cdot 5}{10 \cdot 7} \frac{a^{-4}}{a^{-8}} \frac{b^{-6}}{b^6} = \frac{105}{70} a^{-4-(-8)} b^{-6-6} = \frac{3}{2} a^4 b^{-12} = \frac{3a^4}{2b^{12}} \) - a) \( \left(\frac{9m^{-3}}{5n^{-1}}\right)^{-2} \cdot 81m^{-6}n^3 = \frac{(9m^{-3})^{-2}}{(5n^{-1})^{-2}} \cdot 81m^{-6}n^3 = \frac{9^{-2}(m^{-3})^{-2}}{5^{-2}(n^{-1})^{-2}} \cdot 81m^{-6}n^3 = \frac{9^{-2}m^6}{5^{-2}n^2} \cdot 81m^{-6}n^3 = \frac{1}{81}m^6 \cdot 25n^2 \cdot 81m^{-6}n^3 = (\frac{1}{81} \cdot 81) (m^6 \cdot m^{-6}) (n^2 \cdot n^3) \cdot 25 = 1 \cdot m^0 \cdot n^5 \cdot 25 = 25n^5 \)
б) \( \left(\frac{2x^4}{y^9}\right)^{-3} \cdot (x^{-2}y)^{-6} = \frac{(2x^4)^{-3}}{(y^9)^{-3}} \cdot (x^{-2})^{-6}y^{-6} = \frac{2^{-3}(x^4)^{-3}}{y^{9 \cdot (-3)}} \cdot x^{-2 \cdot (-6)}y^{-6} = \frac{2^{-3}x^{-12}}{y^{-27}} \cdot x^{12}y^{-6} = 2^{-3} \frac{x^{-12}x^{12}}{y^{-27}y^6} = \frac{1}{8} \frac{x^0}{y^{-21}} = \frac{1}{8} y^{21} = \frac{y^{21}}{8} \)
