NHẬN BIẾT (5 câu)
Câu 1: Thực hiện phép tính sau
a)$\frac{x^{2}+y^{2}}{x-y}+\frac{2xy}{y-x}$
b) $\frac{5x-7}{2(x-1)}-\frac{4x}{x^{2}-1}+\frac{9-3x}{2(x-1)}$
Câu 2: Thực hiện các phép tính
a) $\frac{x^{3}}{x+3}+\frac{3x^{2}}{x+3}$
b) $\frac{4}{x^{2}-4x}+\frac{x-8}{4x-16}$
Câu 3: Thực hiện các phép tính
a) $\left ( \frac{3x}{1-3x}+\frac{2x}{3x+1} \right ):\frac{6x^{2}+10x}{1-6x+9x^{2}}$
b) $\frac{a+b}{a^{2}+b^{2}+ab}+\frac{1}{a-b}-3.\frac{ab}{a^{3}-b^{3}}$
Câu 4: Thực hiện các phép tính sau
a) $\frac{4x}{2x-3}-\frac{6}{2x-3}$
b) $\frac{6}{x^{2}+4x}+\frac{3}{2x+8}$
Câu 5: Thực hiện phép tính
a) $\frac{3x-12}{5x^{3}}:\frac{x-4}{15x}$
b) $\frac{x^{2}-3x}{x^{2}+2x+1}:\frac{x-3}{x+1}$
Bài Làm:
Câu 1:
a)$\frac{x^{2}+y^{2}}{x-y}+\frac{2xy}{y-x}$
$=\frac{x^{2}+y^{2}}{x-y}-\frac{2xy}{x-y}=\frac{x^{2}-2xy+y^{2}}{x-y}=\frac{(x-y)^{2}}{x-y}=x-y$
b) $\frac{5x-7}{2(x-1)}-\frac{4x}{x^{2}-1}+\frac{9-3x}{2(x-1)}$
$=\frac{5x-7}{2(x-1)(x+1)}-\frac{4x}{(x-1)(x+1)}+\frac{9-3x}{2(x-1)}$
$=\frac{(5x-7)(x+1)}{2(x-1)(x+1)}-\frac{2.4x}{2.(x-1)(x+1)}+\frac{(9-3x)(x+1)}{2(x-1)(x+1)}$
$=\frac{(5x-7)(x+1)-8x+(9-3x)(x+1)}{2(x-1)(x+1)}$
$=\frac{5x^{2}+5x-7x-7-8x+9x+9-3x^{2}-3x}{2(x-1)(x+1)}$
$=\frac{2x^{2}-4x+2}{2(x-1)(x+1)}=\frac{(x-1)^{2}}{(x-1)(x+1)}=\frac{(x-1)}{(x-1)}$
Câu 2:
a) $\frac{x^{3}}{x+3}+\frac{3x^{2}}{x+3}$
$=\frac{x^{3}+3x^{2}}{x+3}=\frac{x^{2}(x+3)}{x+3}=x^{2}$
b) $\frac{4}{x^{2}-4x}+\frac{x-8}{4x-16}$
$=\frac{4}{x(x-4)}+\frac{x-8}{4(x-4)}$
$=\frac{4.4}{4x(x-4)}+\frac{x(x-8)}{4x(x-4)}=\frac{16+x^{2}-8x}{4x(x-4)}$
$=\frac{(x-4)^{2}}{4x(x-4)}=\frac{x-4}{4x}$
Câu 3:
a) $\left ( \frac{3x}{1-3x}+\frac{2x}{3x+1} \right ):\frac{6x^{2}+10x}{1-6x+9x^{2}}$
$=\frac{3x.(3x+1)+2x(1-3x)}{(1-3x)(1+3x)}:\frac{2x(3x+5)}{(1-3x)^{2}}$
$=\frac{9x^{2}+3x+2x-6x^{2}}{(1-3x)(1+3x)}.\frac{(1-3x)^{2}}{2x.(3x+5)}$
$=\frac{(3x^{2}+5x)(1-3x)}{(1+3x).2x.(3x+5)}$
$=\frac{x.(3x+5)(1-3x)}{2x.(3x+5)(1+3x)}=\frac{1-3x}{2(1+3x)}$
b) $\frac{a+b}{a^{2}+b^{2}+ab}+\frac{1}{a-b}-3.\frac{ab}{a^{3}-b^{3}}$
$=\frac{(a+b)(a-b)}{(a-b)(a^{2}+b^{2}+ab)}+\frac{a^{2}+b^{2}+ab}{(a-b)(a^{2}+b^{2}+ab)}-\frac{3ab}{(a-b)(a^{2}+b^{2}+ab)}$
$=\frac{a^{2}-b^{2}+a^{2}+b^{2}+ab-3ab}{(a-b)(a^{2}+b^{2}+ab)}$
$=\frac{2a^{2}-2ab}{(a-b)(a^{2}+b^{2}+ab)}$
$=\frac{2a(a-b)}{(a-b)(a^{2}+b^{2}+ab)}$
$=\frac{2a}{a^{2}+b^{2}+ab}$
Câu 4:
a) $\frac{4x}{2x-3}-\frac{6}{2x-3}$
$\frac{4x-6}{2x-3}=\frac{2(2x-3)}{2x-3}=2$
b) $\frac{6}{x^{2}+4x}+\frac{3}{2x+8}$
$\frac{6.2}{x(x+4).2}+\frac{3x}{2(x+4).x}$
$=\frac{12+3x}{2x(x+4)}=\frac{3(x+4)}{2x(x+4)}$
$=\frac{3}{2x}$
Câu 5:
a) $\frac{3x-12}{5x^{3}}:\frac{x-4}{15x}$
$=\frac{3x-12}{5x^{3}}.\frac{15}{x-4}$
$=\frac{(3x-12)15x}{5x^{3}(x-4)}=\frac{3(x-4)15x}{5x^{3}(x-4)}$
$=\frac{9}{x^{2}}$
b) $\frac{x^{2}-3x}{x^{2}+2x+1}:\frac{x-3}{x+1}$
$\frac{x^{2}-3x}{x^{2}+2x+1}.\frac{x+1}{x-3}=\frac{x(x-3)(x+1)}{(x+1)^{2}(x-3)}=\frac{x}{x+1}$