2. Вычислите: 1) a) 37-3-6; б) 7-9.78; B) 1 -7 6 1 ; г) 1 14 -16 1 ; 5 5 4 4 -3 -3 2) a) 25:26; б) 5:5-2; в) 6-9:6-11; г) 1 1 ; 7 7 -2 -3 3) a) (32)-1; б) 1 ; в) (0,1-2)6; г) 1 ; 2 6 -2 4) a) -17-34-1; б) -10·2-3; B) -0,01-1; г) 6-2 + 24-1; 1 8 -6 5) a) 32.2-6; 6) 27-(3-2)2; B) 7-8.79 :49; г) 25-2; 5 a) 81-2-272; б) 16-5 : 8-6; B) (-6)-9.6-7 4-6.16-5 ; г) 6-15 8-10 3. Упростите выражение: a) 6x-5y72,5x7y-6; б) 0,8а-6645a126-4; a) 3,2a6b: (0,8a3b-³); 1-8 б) 3-m-n-7: 7 m-5n-7; 2 8 б) 21a-4 56-6 1066 7-8; a -3 6) 2x4 (x-2y)-6.

2. Вычислите:

1)

а) 3⁷ ⋅ 3^-6 = 3^7+(-6) = 3¹ = 3

б) 7^-9 ⋅ 7⁸ = 7^-9+8 = 7^-1 = \(\frac{1}{7}\)

в) (\(\frac{1}{5}\))^-7 ⋅ (\(\frac{1}{5}\))⁶ = (\(\frac{1}{5}\))^-7+6 = (\(\frac{1}{5}\))^-1 = 5

г) (\(\frac{1}{4}\))¹⁴ ⋅ (\(\frac{1}{4}\))^-16 = (\(\frac{1}{4}\))^14+(-16) = (\(\frac{1}{4}\))^-2 = 4² = 16

2)

а) 2⁵ : 2⁶ = 2^5-6 = 2^-1 = \(\frac{1}{2}\)

б) 5 : 5^-2 = 5^1-(-2) = 5¹⁺² = 5³ = 125

в) 6^-9 : 6^-11 = 6^-9-(-11) = 6^-9+11 = 6² = 36

г) (\(\frac{1}{7}\))^-3 : (\(\frac{1}{7}\))^-3 = (\(\frac{1}{7}\))^-3-(-3) = (\(\frac{1}{7}\))^-3+3 = (\(\frac{1}{7}\))⁰ = 1

3)

а) (3²)^-1 = 3^2⋅(-1) = 3^-2 = \(\frac{1}{3^2}\) = \(\frac{1}{9}\)

б) ((\( \frac{1}{2} \))^-3)^-2 = (\(\frac{1}{2}\))^-3⋅(-2) = (\(\frac{1}{2}\))⁶ = \(\frac{1}{2^6}\) = \(\frac{1}{64}\)

в) (0,1^-2)⁶ = 0,1^-2⋅6 = 0,1^-12 = (\(\frac{1}{10}\))^-12 = 10¹²

г) ((\(\frac{1}{6}\))^-2)⁰ = (\(\frac{1}{6}\))^-2⋅0 = (\(\frac{1}{6}\))⁰ = 1

4)

а) -17 ⋅ 34^-1 = -17 ⋅ \(\frac{1}{34}\) = -\(\frac{17}{34}\) = -\(\frac{1}{2}\)

б) -10 ⋅ 2^-3 = -10 ⋅ \(\frac{1}{2^3}\) = -10 ⋅ \(\frac{1}{8}\) = -\(\frac{10}{8}\) = -\(\frac{5}{4}\)

в) (\(\frac{1}{8}\))^-2 - 0,01^-1 = 8² - (\(\frac{1}{100}\))^-1 = 64 - 100 = -36

г) 6^-2 + 24^-1 = \(\frac{1}{6^2}\) + \(\frac{1}{24}\) = \(\frac{1}{36}\) + \(\frac{1}{24}\) = \(\frac{2}{72}\) + \(\frac{3}{72}\) = \(\frac{5}{72}\)

5)

а) 32 ⋅ 2^-6 = 2⁵ ⋅ 2^-6 = 2^5+(-6) = 2^-1 = \(\frac{1}{2}\)

б) 27 ⋅ (3^-2)² = 3³ ⋅ 3^-4 = 3^3+(-4) = 3^-1 = \(\frac{1}{3}\)

в) 7^-8 ⋅ 7⁹ : 49 = 7^-8 ⋅ 7⁹ : 7² = 7^-8+9-2 = 7^-1 = \(\frac{1}{7}\)

г) 25^-2 ⋅ (\(\frac{1}{5}\))^-6 = (5²)^-2 ⋅ (\(\frac{1}{5}\))^-6 = 5^-4 ⋅ 5⁶ = 5^-4+6 = 5² = 25

6)

а) 81^-2 ⋅ 27² = (3⁴)^-2 ⋅ (3³)² = 3^-8 ⋅ 3⁶ = 3^-8+6 = 3^-2 = \(\frac{1}{3^2}\) = \(\frac{1}{9}\)

б) 16^-5 : 8^-6 = (2⁴)^-5 : (2³)^-6 = 2^-20 : 2^-18 = 2^-20-(-18) = 2^-20+18 = 2^-2 = \(\frac{1}{2^2}\) = \(\frac{1}{4}\)

в) \(\frac{(-6)^{-9} ⋅ 6^{-7}}{6^{-15}}\) = \(\frac{(-1)^{-9} ⋅ 6^{-9} ⋅ 6^{-7}}{6^{-15}}\) = \(\frac{-1 ⋅ 6^{-16}}{6^{-15}}\) = -6^-16-(-15) = -6^-16+15 = -6^-1 = -\(\frac{1}{6}\)

г) \(\frac{4^{-6} ⋅ 16^{-5}}{8^{-10}}\) = \(\frac{(2^2)^{-6} ⋅ (2^4)^{-5}}{(2^3)^{-10}}\) = \(\frac{2^{-12} ⋅ 2^{-20}}{2^{-30}}\) = \(\frac{2^{-32}}{2^{-30}}\) = 2^{-32-(-30)} = 2^{-32+30} = 2^{-2} = \(\frac{1}{2^2}\) = \(\frac{1}{4}\)

3. Упростите выражение:

a)

6x^-5y⁷ ⋅ 2,5x⁷y^-6 = 6 ⋅ 2,5 ⋅ x^-5+7 ⋅ y^7+(-6) = 15x²y

б)

0,8а^-6b⁴ ⋅ 5a¹²b^-4 = 0,8 ⋅ 5 ⋅ a^-6+12 ⋅ b^4+(-4) = 4a⁶

a)

3,2a⁶b : (0,8a³b^-3) = (3,2 : 0,8) ⋅ a^6-3 ⋅ b^1-(-3) = 4a³b⁴

б)

3\(\frac{1}{2}\)m^-8n^-7 : (\(\frac{7}{8}\)m^-5n^-7) = \(\frac{7}{2}\)m^-8n^-7 ⋅ \(\frac{8}{7}\)m⁵n⁷ = (\(\frac{7}{2}\) ⋅ \(\frac{8}{7}\)) ⋅ m^-8+5 ⋅ n^-7+7 = 4m^-3

б)

\(\frac{21a^{-4}}{10b^6}\) ⋅ \(\frac{5b^{-6}}{7a^{-8}}\) = \(\frac{21 ⋅ 5}{10 ⋅ 7}\) ⋅ \(\frac{a^{-4}}{a^{-8}}\) ⋅ \(\frac{b^{-6}}{b^6}\) = \(\frac{3}{1}\) ⋅ \(\frac{1}{2}\) ⋅ a^-4-(-8) ⋅ b^-6-6 = \(\frac{3}{2}\)a⁴b^-12

б)

(( \(\frac{2x^4}{y^9}\) )^-3 ⋅ (x^-2y)^-6 = (\(\frac{2^{-3} ⋅ x^{-12}}{y^{-27}}\) ⋅ x¹²y^-6 = \(\frac{1}{8}\) ⋅ x^-12+12 ⋅ y^-6+27 = \(\frac{1}{8}\)y²¹