1.
числитель:
а²/(а+в) - а³/(а²+2ав+в²) = а²/(а+в) - а³/(а+в)² =
= а²(а+в)/(а+в)² - а³/(а+в)² = (а³+а²в-а³)/(а+в)² = а²в/(а+в)²,
знаменатель:
а/(а+в) - а²/(а²-в²) = а/(а+в) - а²/(а-в)(а+в) =
= а(а-в)/(а-в)(а+в) - а²/(а-в)(а+в) = (а²-ав-а²)/(а-в)(а+в) = -ав/(а-в)(а+в),
значение дроби:
а²в/(а+в)² : -ав/(а-в)(а+в) = а²в/(а+в)² * (а-в)(а+в)/(-ав) = -а(а-в)/(а+в),
2.
скобки:
z/(2z-4) - (z²+4)/(2z²-8) - 2/(z²+2z) =
= z/2*(z-2) - (z²+4)/2*(z-2)(z+2) - 2/z*(z+2) =
= [ z*z(z+2) - z*(z²+4) - 2*2(z-2) ] / (2z(z-2)(z+2)) =
= [z³+2z² - z³-4z - 4z+8] / (2z(z-2)(z+2)) =
= (2z²-8z+8) / (2z(z-2)(z+2)) =
= 2(z²-4z+4) / (2z(z-2)(z+2)) =
= (z²-4z+4) / (z(z-2)(z+2)) =
= (z-2)² / (z(z-2)(z+2) = (z-2) / (z(z+2)),
деление:
(z-2)/(4z²+16z+16) : (z-2) / (z(z+2)) = (z-2)/(4*(z+2)² * (z(z+2)) / (z-2) =
= z/(4*(z+2))
3.
2/х - (х-2)/(х²-х) = 2/х - (х-2)/(х*(х-1)) = 2(х-1)/(х*(х-1)) - (х-2)/(х*(х-1)) =
= (2х-2 - х+2) / ((х*(х-1)) = х/(х*(х-1)) = 1/(х-1),
3/х + (х+3)/(х²-х) = 3/х - (х+3)/(х*(х-1)) = 3(х-1)/(х*(х-1)) - (х+3)/(х*(х-1)) =
= (3х-3 - х-3) / ((х*(х-1)) = (2х-6)/(х*(х-1)) = 2*(х-3)/(х*(х-1)),
1/(х-1) : 2*(х-3)/(х*(х-1)) = 1/(х-1)* х*(х-1)/2*(х-3) = х/(2(х-3)),
4.
(а+5)/(5а-1) + (а+5)/(а+1) =
= (а+5)(а+1)/(5а-1)(а+1) + (а+5)(5а-1)/(5а-1)(а+1) =
= (а²+а+5а+5 + 5а²-а +25а-5) / (5а-1)(а+1) =
= (6а²+30а) / (5а-1)(а+1) =
= 6а(а+5)/(5а-1)(а+1),
6а(а+5)/(5а-1)(а+1) : (а²+5а)/(1-5а) =
= 6а(а+5)/(5а-1)(а+1) * (1-5а)/(а*(а+5)) = -6/(а+1),
сложение:
-6/(а+1) + (а²+5)/(а+1) = (-6+а²+5)/(а+1) =
= (а²-1)/(а+1) = (а-1)(а+1)/(а+1) = а-1
=
1. Воспользуемся следующими тригонометрическими формулами:
sina + sinb = 2sin((a + b)/2) * cos((a - b)/2);
cos2a = 1 - 2sin^2(a);
sin3x + sin5x + 2sin^2(x/2) = 1;
2sin((5x + 3x)/2) * cos((5x - 3x)/2) - (1 - 2sin^2(x/2)) = 0;
2sin4x * cosx - cosx = 0.
2. Вынесем общий множитель cosx за скобки:
cosx(2sin4x - 1) = 0;
[cosx = 0;
[2sin4x - 1 = 0;
[sin4x = 1/2;
[x = π/2 + πk, k ∈ Z;
[4x = π/6 + 2πk; 5π/6 + 2πk, k ∈ Z;
[x = π/24 + πk/2; 5π/24 + πk/2, k ∈ Z.
ответ: π/2 + πk; π/24 + πk/2; 5π/24 + πk/2, k ∈ Z.
1.
числитель:
а²/(а+в) - а³/(а²+2ав+в²) = а²/(а+в) - а³/(а+в)² =
= а²(а+в)/(а+в)² - а³/(а+в)² = (а³+а²в-а³)/(а+в)² = а²в/(а+в)²,
знаменатель:
а/(а+в) - а²/(а²-в²) = а/(а+в) - а²/(а-в)(а+в) =
= а(а-в)/(а-в)(а+в) - а²/(а-в)(а+в) = (а²-ав-а²)/(а-в)(а+в) = -ав/(а-в)(а+в),
значение дроби:
а²в/(а+в)² : -ав/(а-в)(а+в) = а²в/(а+в)² * (а-в)(а+в)/(-ав) = -а(а-в)/(а+в),
2.
скобки:
z/(2z-4) - (z²+4)/(2z²-8) - 2/(z²+2z) =
= z/2*(z-2) - (z²+4)/2*(z-2)(z+2) - 2/z*(z+2) =
= [ z*z(z+2) - z*(z²+4) - 2*2(z-2) ] / (2z(z-2)(z+2)) =
= [z³+2z² - z³-4z - 4z+8] / (2z(z-2)(z+2)) =
= (2z²-8z+8) / (2z(z-2)(z+2)) =
= 2(z²-4z+4) / (2z(z-2)(z+2)) =
= (z²-4z+4) / (z(z-2)(z+2)) =
= (z-2)² / (z(z-2)(z+2) = (z-2) / (z(z+2)),
деление:
(z-2)/(4z²+16z+16) : (z-2) / (z(z+2)) = (z-2)/(4*(z+2)² * (z(z+2)) / (z-2) =
= z/(4*(z+2))
3.
числитель:
2/х - (х-2)/(х²-х) = 2/х - (х-2)/(х*(х-1)) = 2(х-1)/(х*(х-1)) - (х-2)/(х*(х-1)) =
= (2х-2 - х+2) / ((х*(х-1)) = х/(х*(х-1)) = 1/(х-1),
знаменатель:
3/х + (х+3)/(х²-х) = 3/х - (х+3)/(х*(х-1)) = 3(х-1)/(х*(х-1)) - (х+3)/(х*(х-1)) =
= (3х-3 - х-3) / ((х*(х-1)) = (2х-6)/(х*(х-1)) = 2*(х-3)/(х*(х-1)),
значение дроби:
1/(х-1) : 2*(х-3)/(х*(х-1)) = 1/(х-1)* х*(х-1)/2*(х-3) = х/(2(х-3)),
4.
скобки:
(а+5)/(5а-1) + (а+5)/(а+1) =
= (а+5)(а+1)/(5а-1)(а+1) + (а+5)(5а-1)/(5а-1)(а+1) =
= (а²+а+5а+5 + 5а²-а +25а-5) / (5а-1)(а+1) =
= (6а²+30а) / (5а-1)(а+1) =
= 6а(а+5)/(5а-1)(а+1),
деление:
6а(а+5)/(5а-1)(а+1) : (а²+5а)/(1-5а) =
= 6а(а+5)/(5а-1)(а+1) * (1-5а)/(а*(а+5)) = -6/(а+1),
сложение:
-6/(а+1) + (а²+5)/(а+1) = (-6+а²+5)/(а+1) =
= (а²-1)/(а+1) = (а-1)(а+1)/(а+1) = а-1
=
1. Воспользуемся следующими тригонометрическими формулами:
sina + sinb = 2sin((a + b)/2) * cos((a - b)/2);
cos2a = 1 - 2sin^2(a);
sin3x + sin5x + 2sin^2(x/2) = 1;
2sin((5x + 3x)/2) * cos((5x - 3x)/2) - (1 - 2sin^2(x/2)) = 0;
2sin4x * cosx - cosx = 0.
2. Вынесем общий множитель cosx за скобки:
cosx(2sin4x - 1) = 0;
[cosx = 0;
[2sin4x - 1 = 0;
[cosx = 0;
[sin4x = 1/2;
[x = π/2 + πk, k ∈ Z;
[4x = π/6 + 2πk; 5π/6 + 2πk, k ∈ Z;
[x = π/2 + πk, k ∈ Z;
[x = π/24 + πk/2; 5π/24 + πk/2, k ∈ Z.
ответ: π/2 + πk; π/24 + πk/2; 5π/24 + πk/2, k ∈ Z.