2. Вычислите:
1) a) 3^7 * 3^-6; б) 7^-9 * 7^8;
2) a) 25^5 : 2^6; б) 5^5 : 5^-2;
3) a) (3^2)^-1; б) ((1/2)^-3)^-2; в) (0.1^-2)^6; г) ((1/6)^-2)^0;
4) a) -17 * 34^-1; б) -10 * 2^-3; в) (1/8)^-2 - 0.01^-1;
г) 6^-2 + 24^-1;
5) a) 32^-2 * 2^-6; б) 27 * (3^-2)^2; в) 7^-8 * 7^9 : 49;
г) 25^-2 * (1/5)^-6;
6) a) 81^-2 * 27^2; б) 16^-5 : 8^-6; в) (-6)^-9 * 6^-7 / 6^-15;
4^-6 * 16^-5
г) ------------
8^-10
3. Упростите выражение:
1) a) 6x^-5y^7 * 2.5x^7y^-6; б) 0.8a^-6b^4 * 5a^12b^-4;
2) a) 3,2a^6b : (0,8a^3b^-3); б) 3/2 m^-8n^-7 : (-7/8 m^-5n^-7);
3) a) (13x^-4) / (y^-6) : (y / (52x^-5)); б) (21a^-4) / (10b^6) : (5b^-6) / (7a^-8);
4) a) ((9m^-3)/(5n^-1))^-2 * 81m^-6n^3; б) ((2x^4)/y^9)^-3 * (x^-2y^-6).
2. Вычисление:
- \(a)\) \(3^7 \cdot 3^{-6} = 3^{7-6} = 3^1 = 3\)
\(б)\) \(7^{-9} \cdot 7^8 = 7^{-9+8} = 7^{-1} = \frac{1}{7}\) - \(a)\) \(25^5 : 2^6\) - данное выражение не упрощается до целого числа.
- \(a)\) \(\frac{(3^2)^{-1}}{1} = \frac{3^{-2}}{1} = \frac{1}{3^2} = \frac{1}{9}\)
\(б)\) \(\left( \left( \frac{1}{2} \right)^{-3} \right)^{-2} = \left( \frac{1}{2} \right)^{(-3) \cdot (-2)} = \left( \frac{1}{2} \right)^6 = \frac{1}{64}\)
\(в)\) \((0.1^{-2})^6 = 0.1^{-12} = (10^{-1})^{-12} = 10^{12}\)
\(г)\) \(\left( \frac{1}{6} \right)^{-2 \cdot 0} = \left( \frac{1}{6} \right)^0 = 1\) - \(a)\) \(-17 \cdot 34^{-1} = -17 \cdot \frac{1}{34} = -17 \cdot \frac{1}{2 \cdot 17} = -\frac{1}{2}\)
\(б)\) \(-10 \cdot 2^{-3} = -10 \cdot \frac{1}{2^3} = -10 \cdot \frac{1}{8} = -\frac{10}{8} = -\frac{5}{4}\)
\(в)\) \(\left( \frac{1}{8} \right)^{-2} - 0.01^{-1} = (8^{-1})^{-2} - (10^{-2})^{-1} = 8^2 - 10^2 = 64 - 100 = -36\)
\(г)\) \(6^{-2} + 24^{-1} = \frac{1}{6^2} + \frac{1}{24} = \frac{1}{36} + \frac{1}{24} = \frac{2}{72} + \frac{3}{72} = \frac{5}{72}\) - \(a)\) \(32^{-2} \cdot 2^{-6} = (2^5)^{-2} \cdot 2^{-6} = 2^{-10} \cdot 2^{-6} = 2^{-16}\)
\(б)\) \(27 \cdot (3^{-2})^2 = 3^3 \cdot 3^{-4} = 3^{3-4} = 3^{-1} = \frac{1}{3}\)
\(в)\) \(\frac{(-6)^{-9} \cdot 6^{-7}}{6^{-15}} = \frac{(-1)^{-9} \cdot 6^{-9} \cdot 6^{-7}}{6^{-15}} = \frac{-1 \cdot 6^{-16}}{6^{-15}} = -1 \cdot 6^{-16 - (-15)} = -1 \cdot 6^{-1} = -\frac{1}{6}\)
\(г)\) \(\frac{4^{-6} \cdot 16^{-5}}{8^{-10}} = \frac{(2^2)^{-6} \cdot (2^4)^{-5}}{(2^3)^{-10}} = \frac{2^{-12} \cdot 2^{-20}}{2^{-30}} = \frac{2^{-32}}{2^{-30}} = 2^{-32 - (-30)} = 2^{-2} = \frac{1}{4}\) - \(a)\) \(81^{-2} \cdot 27^2 = (3^4)^{-2} \cdot (3^3)^2 = 3^{-8} \cdot 3^6 = 3^{-2} = \frac{1}{9}\)
\(б)\) \(16^{-5} : 8^{-6} = (2^4)^{-5} : (2^3)^{-6} = 2^{-20} : 2^{-18} = 2^{-20 - (-18)} = 2^{-2} = \frac{1}{4}\)
3. Упрощение выражений:
- \(a)\) \(6x^{-5}y^7 \cdot 2.5x^7y^{-6} = (6 \cdot 2.5) x^{-5+7} y^{7-6} = 15 x^2 y^1 = 15x^2y\)
\(б)\) \(0.8a^{-6}b^4 \cdot 5a^{12}b^{-4} = (0.8 \cdot 5) a^{-6+12} b^{4-4} = 4 a^6 b^0 = 4a^6\) - \(a)\) \(3.2a^6b : (0.8a^3b^{-3}) = \frac{3.2a^6b}{0.8a^3b^{-3}} = \frac{3.2}{0.8} a^{6-3} b^{1-(-3)} = 4 a^3 b^4\)
\(б)\) \(\frac{3}{2} m^{-8}n^{-7} : \left( -\frac{7}{8} m^{-5}n^{-7} \right) = \frac{3 m^{-8}n^{-7}}{2} \cdot \frac{8}{-7 m^{-5}n^{-7}} = \frac{3 \cdot 8}{-7 \cdot 2} m^{-8-(-5)} n^{-7-(-7)} = \frac{24}{-14} m^{-3} n^0 = -\frac{12}{7} m^{-3} = -\frac{12}{7m^3}\) - \(a)\) \(\frac{13x^{-4}}{y^{-6}} : \frac{y}{52x^{-5}} = \frac{13x^{-4}}{y^{-6}} \cdot \frac{52x^{-5}}{y} = \frac{13 \cdot 52 x^{-4} x^{-5}}{y^{-6} y} = 13 \cdot 52 x^{-9} y^{6-1} = 676 x^{-9} y^5 = \frac{676y^5}{x^9}\)
\(б)\) \(\frac{21a^{-4}}{10b^6} : \frac{5b^{-6}}{7a^{-8}} = \frac{21a^{-4}}{10b^6} \cdot \frac{7a^{-8}}{5b^{-6}} = \frac{21 \cdot 7 a^{-4} a^{-8}}{10 5 b^6 b^{-6}} = \frac{147 a^{-12}}{50 b^0} = \frac{147}{50a^{12}}\)
- \(a)\) \(\left( \frac{9m^{-3}}{5n^{-1}} \right)^{-2} \cdot 81m^{-6}n^3 = \frac{(9m^{-3})^{-2}}{(5n^{-1})^{-2}} \cdot 81m^{-6}n^3 = \frac{9^{-2} m^{6}}{5^{-2} n^{2}} \cdot 81m^{-6}n^3 = \frac{1}{81} m^6 \cdot 25 n^{-2} \cdot 81 m^{-6} n^3 = (\frac{1}{81} \cdot 81) (m^6 m^{-6}) (n^{-2} n^3) \cdot 25 = 1 \cdot m^0 \cdot n^1 \cdot 25 = 25n\)
\(б)\) \(\left\(\frac{2x^4}{y^9} \right\)^{-3} \(\cdot\) (x^{-2}y^{-6}) = \(\frac{(2x^4)^{-3}}{(y^9)^{-3}}\) \(\cdot\) x^{-2}y^{-6} = \(\frac{2^{-3}x^{-12}}{y^{-27}}\) \(\cdot\) x^{-2}y^{-6} = \(\frac{1}{8}\) x^{-12} y^{27} x^{-2} y^{-6} = \(\frac{1}{8}\) x^{-14} y^{21} = \(\frac{y^{21}}{8x^{14}}\)\
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