2. Вычислите: 1) a) 3⁷⋅3⁻⁶; 6) 7⁻⁹⋅7⁸; в) (1/5)⁻⁷⋅(1/5)⁶; г) (1/4)¹⁴⋅(1/4)⁻¹⁶; 2) а) 2⁵:2⁶; 6) 5:5⁻²; в) 6⁻⁹:6⁻¹¹; г) (1/7)⁻³:(1/7)⁻³; 3) a) (3²)⁻¹; 6) ((1/2)⁻³)⁻²; в) (0,1)⁻²,⁶; г) ((1/6)⁻²)⁰; 4) a)-17⋅34⁻¹; 6) -10⋅2⁻³;в) (1/8)⁻²-0,01⁻¹; г) 6⁻²+24⁻¹; 5) a) 32⋅2⁻⁶; 6) 27⋅(3⁻²)²; в) 7⁻⁸⋅7⁹:49; г) 25⁻²⋅(1/5)⁻⁶; 6) a) 81⁻²⋅27²; 6) 16⁻⁵:8⁻⁶; в) ((-6)⁻⁹⋅6⁻⁷)/(6⁻¹⁵); г) (4⁻⁶⋅16⁻⁵)/(8⁻¹⁰) 3. Упростите выражение: 1) a) 6x⁻⁵y⁷⋅2,5x⁷y⁻⁶; б) 0,8a⁻⁶b⁴⋅5a¹²b⁻⁴, 2) a) 3,2a⁶b:(0,8a³b⁻³); б) (3 1/2)m⁻⁸n⁻⁷:((-7/8)m⁻⁵n⁻⁷); 3) a) (13x⁻⁴)/(y⁻⁶)⋅y/(52x⁻⁵); б) (21a⁻⁴⋅5b⁻⁶)/(10b⁶⋅7a⁻⁸) ; 4) a) ((9m⁻³)/(5n⁻¹))⁻²⋅81m⁻⁶n³; б) ((2x⁴)/y⁹)⁻³⋅(x⁻²y)⁻⁶ 4. Упростите выражение (n — целое число): а) 14ⁿ/(2ⁿ⁻²⋅7ⁿ); в) (x⁶ⁿyⁿ⁺³)/(x²ⁿyⁿ⁺⁴); д) (6ⁿ⁺¹+6ⁿ⁺³)/37; б) 36ⁿ⁺¹/(6²ⁿ⁺¹); г) (a⁻²ⁿ+aⁿ)/(a⁻ⁿ); е) (5ⁿ+1)/(5⁻ⁿ+1) 5. Сократите дробь: a) (a⁷+a¹³)/(a⁻³+a³); б) (x⁴+2x⁶+x⁷)/(2+x+x⁻²)

2. Вычислите:

1. а) 3⁷⋅3⁻⁶ = 3^(7-6) = 3¹ = 3;
   б) 7⁻⁹⋅7⁸ = 7^(-9+8) = 7⁻¹ = 1/7;
   в) (1/5)⁻⁷⋅(1/5)⁶ = (1/5)^(-7+6) = (1/5)⁻¹ = 5;
   г) (1/4)¹⁴⋅(1/4)⁻¹⁶ = (1/4)^(14-16) = (1/4)⁻² = 4² = 16.
2. а) 2⁵:2⁶ = 2^(5-6) = 2⁻¹ = 1/2;
   б) 5:5⁻² = 5^(1-(-2)) = 5³ = 125;
   в) 6⁻⁹:6⁻¹¹ = 6^(-9-(-11)) = 6² = 36;
   г) (1/7)⁻³:(1/7)⁻³ = (1/7)^(-3-(-3)) = (1/7)⁰ = 1.
3. а) (3²)⁻¹ = 3^(2⋅(-1)) = 3⁻² = 1/9;
   б) ((1/2)⁻³)⁻² = (1/2)^(-3⋅(-2)) = (1/2)⁶ = 1/64;
   в) (0,1)⁻²,⁶ = (1/10)^(-2.6) = (10)²,⁶ = 10^(13/5) = 10^(2.6)
   г) ((1/6)⁻²)⁰ = (1/6)^(-2⋅0) = (1/6)⁰ = 1.
4. а) -17⋅34⁻¹ = -17/34 = -1/2 = -0.5;
   б) -10⋅2⁻³ = -10/2³ = -10/8 = -5/4 = -1.25;
   в) (1/8)⁻²-0,01⁻¹ = 8² - 100 = 64 - 100 = -36;
   г) 6⁻²+24⁻¹ = 1/6² + 1/24 = 1/36 + 1/24 = 2/72 + 3/72 = 5/72.
5. а) 32⋅2⁻⁶ = 32/2⁶ = 2⁵/2⁶ = 1/2;
   б) 27⋅(3⁻²)² = 27⋅(1/3²)² = 27⋅(1/9)² = 27/81 = 1/3;
   в) 7⁻⁸⋅7⁹:49 = 7^(-8+9)/49 = 7/49 = 1/7;
   г) 25⁻²⋅(1/5)⁻⁶ = (1/25)²⋅5⁶ = (1/5²)²⋅5⁶ = 5⁶/5⁴ = 5² = 25.
6. а) 81⁻²⋅27² = (3⁴)⁻²⋅(3³)2 = 3^(-8)⋅3⁶ = 3^(-8+6) = 3⁻² = 1/9;
   б) 16⁻⁵:8⁻⁶ = (2⁴)⁻⁵/(2³)⁻⁶ = 2^(-20)/2^(-18) = 2^(-20+18) = 2⁻² = 1/4;
   в) ((-6)⁻⁹⋅6⁻⁷)/(6⁻¹⁵) = ((-1)⁻⁹⋅6⁻⁹⋅6⁻⁷)/6⁻¹⁵ = (-1⋅6^(-9-7))/6⁻¹⁵ = -6^(-16)/6⁻¹⁵ = -6^(-16+15) = -6⁻¹ = -1/6;
   г) (4⁻⁶⋅16⁻⁵)/(8⁻¹⁰) = ((2²)⁻⁶⋅(2⁴)⁻⁵)/(2³)⁻¹⁰ = (2^(-12)⋅2^(-20))/2^(-30) = 2^(-32)/2^(-30) = 2^(-32+30) = 2⁻² = 1/4.

3. Упростите выражение:

1. а) 6x⁻⁵y⁷⋅2,5x⁷y⁻⁶ = 6⋅2,5⋅x^(-5+7)⋅y^(7-6) = 15x²y;
   б) 0,8a⁻⁶b⁴⋅5a¹²b⁻⁴ = 0,8⋅5⋅a^(-6+12)⋅b^(4-4) = 4a⁶.
2. а) 3,2a⁶b:(0,8a³b⁻³) = (3,2/0,8)⋅a^(6-3)⋅b^(1-(-3)) = 4a³b⁴;
   б) (3 1/2)m⁻⁸n⁻⁷:((-7/8)m⁻⁵n⁻⁷) = (7/2)/(-7/8)⋅m^(-8-(-5))⋅n^(-7-(-7)) = -4m⁻³.
3. а) (13x⁻⁴)/(y⁻⁶)⋅y/(52x⁻⁵) = (13/52)⋅x^(-4-(-5))⋅y^(1-(-6)) = (1/4)xy⁷.
   б) (21a⁻⁴⋅5b⁻⁶)/(10b⁶⋅7a⁻⁸) = (21⋅5)/(10⋅7)⋅a^(-4-(-8))⋅b^(-6-6) = (3/2)a⁴b⁻¹².
4. а) ((9m⁻³)/(5n⁻¹))⁻²⋅81m⁻⁶n³ = (9m⁻³/5n⁻¹)^(-2)⋅81m⁻⁶n³ = (5n⁻¹)²/(9m⁻³)²⋅81m⁻⁶n³ = (25n⁻²)/(81m⁻⁶)⋅81m⁻⁶n³ = 25n;
   б) ((2x⁴)/y⁹)⁻³⋅(x⁻²y)⁻⁶ = (2x⁴)⁻³/(y⁹)⁻³⋅x^(12)⋅y^(-6) = (2⁻³⋅x^(-12))/y^(-27)⋅x^(12)⋅y^(-6) = x⁰⋅y^(27-6)/2³ = y²¹/8.

4. Упростите выражение (n — целое число):

1. а) 14ⁿ/(2ⁿ⁻²⋅7ⁿ) = (2⋅7)ⁿ/(2ⁿ⁻²⋅7ⁿ) = 2ⁿ⋅7ⁿ/(2ⁿ⁻²⋅7ⁿ) = 2^(n-(n-2)) = 2² = 4
2. в) (x⁶ⁿyⁿ⁺³)/(x²ⁿyⁿ⁺⁴) = x^(6n-2n)⋅y^(n+3-(n+4)) = x^(4n)⋅y⁻¹ = (x^(4n))/y;
3. д) (6ⁿ⁺¹+6ⁿ⁺³)/37 = (6ⁿ⁺¹(1+6²))/37 = (6ⁿ⁺¹⋅(1+36))/37 = (6ⁿ⁺¹⋅37)/37 = 6ⁿ⁺¹;
4. б) 36ⁿ⁺¹/(6²ⁿ⁺¹) = (6²)ⁿ⁺¹/(6²ⁿ⁺¹) = (6^(2n+2))/(6^(2n+1)) = 6^(2n+2-(2n+1)) = 6¹ = 6;
5. г) (a⁻²ⁿ+aⁿ)/(a⁻ⁿ) = a⁻²ⁿ/a⁻ⁿ + aⁿ/a⁻ⁿ = a^(-2n+n) + a^(n+n) = a⁻ⁿ + a^(2n) = 1/aⁿ + a^(2n);
6. е) (5ⁿ+1)/(5⁻ⁿ+1) = (5ⁿ+1)/(1/5ⁿ+1) = (5ⁿ+1)/((1+5ⁿ)/5ⁿ) = (5ⁿ(5ⁿ+1))/(5ⁿ+1) = 5ⁿ

5. Сократите дробь:

1. а) (a⁷+a¹³)/(a⁻³+a³) = a⁷(1+a⁶)/(a⁻³(1+a⁶)) = a⁷/a⁻³ = a^(7+3) = a¹⁰;
2. б) (x⁴+2x⁶+x⁷)/(2+x+x⁻²) = x⁴(1+2x²+x³)/(x⁻²(2x²+x³+1)) = x⁴⋅x² = x⁶;