Вправи: 180

180. Спростіть вираз:

1) (15x-7 – x – 7) • 7-xx2- 16x+64 = x+8x-8.

1. 15x-7 – (x + 7) = 15x-7 – x+ 7\x-71 = 15-x-7(x+7)x-7 = 15-(x2- 49)x-7 = 15- x2+ 49x-7 = 64- x2x-7;

2. 8-x(8+x) x-7 • 7-x(x-8)2 = -x-8x+7•(-x-7)x-7(x-8)2 = x+8x-8;

2) (a – 5a-16a-3) : (2a – 2aa-3) = a-42a.

1. a\a-31 – 5a-16a-3 = aa-3- (5a-16)a-3 = a2- 3a-5a+16a-3 = a2- 8a+16a-3 = (a-4)2a-3;

2. 2aa-31 – 2aa-3 = 2aa-3- 2aa-3 = 2a2- 6a-2aa-3 = 2a2- 8aa-3 = 2a(a-4)a-3;

3. (a-4)2a-3 : 2a(a-4)a-3 = (a-4)2•(a-3)a-3•2a(a-4) = a-42a;

3) (1a + 2b + ab2) • aba2- b2 + 2b-a = 1b.

1. 1\b2a + 2\abb + aab2 = b2+ 2ab+ a2ab2 = (a+b)2ab2;

2. (a+b)2ab2 • aba-b(a+b) = a+b • abab2a-b(a+b) = a+bb(a-b);

3. a+bb(a-b) – 2\ba-b = a+b-2bb(a-b) = a-bb(a-b) = ab;

4) (aa-1 – a a+1 – a2+ 11- a2) : a2+ a(a-1)2 = a-1a.

1. a\a+1a-1 – a\a-1a+1 + a2+ 1a-1(1+a) = aa+1- aa-1+ a2+ 1a-1(1+a) = a2+ a- a2+ a+ a2+ 1a-1(1+a) = a2+ 2a+1a-1(a+1) = (a+1)2a-1(a+1) = a+1a-1;

2. a+1a-1 : a2+ a(a-1)2 = (a+1(a-1)2a-1 • a(a+1) = a-1a;

5) (x+2y\x+2yx-2y – x-2y\x-2yx+2y – 16y2x2- 4y2) : 4yx+2y = 2.

1. x+2y\x+2yx-2y – x-2y\x-2yx+2y – 16y2x2- 4y2=(x+2y)2- (x-2y)2- 16y2x-2y(x+2y) = x2+ 4xy+4y2- x2+ 4xy-4y2- 16y2x2- 4y2 • x+2y4y = x2+ 4xy+4y2- x2- 4xy+4y2- 16y2x-2y(x+2y) = x2+ 4xy+4y2- x2+ 4xy-4y2- 16y2x-2y(x+2y) = 8xy-16y2x-2y(x+2y) = 8y(x-2y)x-2y(x+2y) = 8yx+2y;

2. 8yx+2y : 4yx+2y = 8y • (x+2y)x+2y • 4y = 2;

6) (3a-8a2- 2a+4 + 1a+2 – 4a-28a3+ 8) • a2- 44 = a – 2.

1. 3a- 8\a+2a2- 2a+4 + 1\a2-2a+4a+2 – 4a-28a+2(a2- 2a+4) = a+23a-8+ a2- 2a+4-(4a-28)a+2(a2- 2a+4) = 3a2- 8a+6a-16+ a2- 2a+4-4a+28a+2(a2- 2a+4) = 4a2- 8a+16a+2(a2- 2a+4) = 4(a2- 2a+4)a+2(a2- 2a+4) = 4a+2;

2. 4a+2 • a-2(a+2)4 = a – 2.