Вправи: 403 (1-6)

403. Спростіть вираз:

1) a2a+b – b²(a + b)^–1 = a2a+b – b2a+b = a2- b2a+b = a-b(a+b)a+b = a – b;

2) 1x+y – x(x + y)^–2 = 1\x+yx+y – x(x+y)2 = x+y-x(x+y)2 = y(x+y)2;

3) 2m3m+1 – 2m³(1 – m²)^–1 = 2m3m+1 – 2m31-m2 = 2m3\1+m1+m – 2m31-m(1+m) = 2m3+ 2m3+ 2m4- 2m31-m(1+m) = 2m41-m2;

4) 4(a – 2)^–1 – a2a-2 = 4a-2 – a2a-2 = 4-a2a-2 = 2-a(2+a)a-2 = –a-2(a+2)a-2 = a – 2

5) (x – 5)^–1 + x(x – 5)^–1 – 6x+5 = 1x-5 + xx-5 –

6) (x + y)^–2 – (x – y)^–2 = 1\(x-y)2(x+y)2 – 1\(x+y)2(x-y)2 = x2- 2xy+ y2- (x2+ 2xy+ y2) (x+y)2(x-y)2 = x2- 2xy+ y2- x2- 2xy- y2(x+y)2 • (x-y)2 = –4xy(x+y)2(x-y)2 ;

7) (1 + b)^–1 – (1 + b)^–2 = 1\1+b1+b – 1(1+b)2 = 1+b -1(1+b)2 = b(1+b)2;

8) (m + n)^–1 – (m + n)^–2 = 1\m+nm+n – 1(m+n)2 = m+n-1(m+n)2;

9) mn(m² – 1)^–1 – (m-1n)^–1 = mnm2- 1 – nm-1 = mnm-1m+1 – n\m+1m-1 = mn-mn-nm-1(m+1) = –nm2- 1 = n1-m2;

10) 1 – (1 + x^–1)^–1 = 1 – 11+ 1x = 1 – 1x+1x = 1^\x+1 – xx+1 = x+1-xx+1 = 1x+1;

11) (1 + (1 + a^–1)^–1)^–1 = (1 + (1 + 1a)^–1)^–1 = (1 + (a+1a)^–1)^–1 = (1 + aa+1)^–1 = (a+1+aa+1)^–1 = (2a+1a+1)^–1 = a+12a+1;

12) (a – a^–1)^–2 + (1 – a²)^–2 = (a – 1a)^–2 + 1(1-a2)2 = (a2- 1a)^–2 + 1(1-a2)2 = (aa2- 1)² + 1(a2- 1)2 = a2(a2- 1)2 + 1(a2- 1)2 = a2+ 1(a2- 1)2.