Đề thi online - Nhân, chia các phân thức đại số - Có lời giải chi tiết

Câu 1 : Thực hiện phép tính \(\frac{{3x + 15}}{{{x^2} - 4}}\,\,:\,\,\frac{{x + 5}}{{x - 2}}\) ta được:

A \(\frac{{3(x - 2)}}{{x + 2}}\)

B \(\frac{{3(x + 5)}}{{x - 2}}\)

C \(\frac{3}{{x - 2}}\)

D \(\frac{3}{{x + 2}}\)

Câu 2 : Thực hiện phép tính: a) \(\frac{{3{x^2} - 6xy + 3{y^2}}}{{5{x^2} - 5xy + 5{y^2}}}\,\,\,:\,\,\frac{{10x - 10y}}{{{x^3} + {y^3}}}\)b) \(\frac{{x - 6}}{{{x^2} + 1}}\,\, \cdot \,\,\frac{{3{x^2} - 3x + 3}}{{{x^2} - 36}} + \frac{{x - 6}}{{{x^2} + 1}}\,\, \cdot \,\,\frac{{3x}}{{{x^2} - 36}}\)c) \(\frac{{x - 1}}{{{x^2} - 4x + 4}}\, \cdot \,\frac{{{x^2} - 4}}{{{x^3} - 1}}\, \cdot \,\frac{{{x^2} + x + 1}}{{x + 2}}\)d) \(\left( {\frac{{3x}}{{1 - 3x}} + \frac{{2x}}{{3x + 1}}} \right):\frac{{6{x^2} + 10x}}{{1 - 6x + 9{x^2}}}\)

A a) \(\frac{{3({x^2} - {y^2})}}{{50}}.\)

b) \(\frac{1}{x - 2}.\)

c) \(\frac{3}{{x + 6}}.\)

d) \(\frac{{1 - 3x}}{{2(3x + 1)}}.\)

B a) \(\frac{{3({x} - {y})}}{{50}}.\)

b) \(\frac{1}{x +2}.\)

c) \(\frac{3}{{x - 6}}.\)

d) \(\frac{{3x}}{{2(3x + 1)}}.\)

C a) \(\frac{{3({x} - {y})}}{{5}}.\)

b) \(\frac{3}{x +2}.\)

c) \(\frac{15}{{x + 6}}.\)

d) \(\frac{{3x}^2}{{2(3x + 1)}}.\)

D a) \(\frac{{3({x}^3 - {y}^3)}}{{50}}.\)

b) \(\frac{6}{x +2}.\)

c) \(\frac{3}{{x - 6}^2}.\)

d) \(\frac{{3x}}{{(3x - 1)}}.\)

Câu 3 : Rút gọn biểu thức: a) \(\left( {\frac{9}{{{x^3} - 9x}} + \frac{1}{{x + 3}}} \right):\left( {\frac{{x - 3}}{{{x^2} + 3x}} - \frac{x}{{3x + 9}}} \right)\)b) \(\frac{{x - 2}}{{{x^2} + 3x + 2}}\, \cdot \,\frac{{x + 2}}{{{x^2} - 5x + 6}}\)c) \(\frac{{{x^2} + 1}}{{3x}}:\frac{{{x^2} + 1}}{{x - 1}}:\frac{{{x^3} - 1}}{{{x^2} + x}}:\frac{{{x^2} + 2x + 1}}{{{x^2} + x + 1}}\)d) \(\frac{{x + 3}}{{{x^2} - 1}}:\frac{{x + 4}}{{{x^2} + 6x}} - \frac{{x + 3}}{{{x^2} - 1}}:\frac{{x + 4}}{{x - 4}}\)

A a) \(\frac{{ 3}}{{x - 3}}.\)

b) \(\frac{-1}{{(x + 1)(x - 3)}}.\)

c) \(\frac{1}{{3(x - 1)}}.\)

d) \(\frac{{x - 3}}{{x - 1}}.\)

B a) \(\frac{{ - 3}}{{x - 3}}.\)

b) \(\frac{1}{{(x + 1)(x - 3)}}.\)

c) \(\frac{1}{{3(x + 1)}}.\)

d) \(\frac{{x + 3}}{{x - 1}}.\)

C a) \(\frac{{ - 3}}{{x + 3}}.\)

b) \(\frac{1}{{(x -1)(x +3)}}.\)

c) \(\frac{1}{{3(x -1)}}.\)

d) \(\frac{{x -3}}{{x - 1}}.\)

D a) \(\frac{{ 4}}{{x - 3}}.\)

b) \(\frac{1}{{(x -1)(x - 3)}}.\)

c) \(\frac{-1}{{3(x -1)}}.\)

d) \(\frac{{x + 3}}{{x + 1}}.\)

Câu 4 : Tìm biểu thức M, biết:a) \(\frac{{x + 2y}}{{{x^3} - 8{y^3}}}\, \cdot \,M = \frac{{5{x^2} + 10xy}}{{{x^2} + 2xy + 4{y^2}}}\)b) \(M:\frac{{{x^2} + x + 1}}{{2x + 2}} = \frac{{x + 1}}{{{x^3} - 1}}\)c) \(\frac{{{x^2} + xy - 2{y^2}}}{{{x^4} - {y^4}}}\, \cdot \,M = \frac{{x + y}}{{{x^3} + {x^2}y + x{y^2} + {y^3}}}\)

A a) \(M = \frac{{5{x} + 10xy}}{{{x} + 2xy + 4{y^2}}}.\)

b) \(M = \frac{1}{{(x - 1)}}.\)

c) \(M = \frac{{x + y}}{{x - 2y}}.\)

B a) \(M = \frac{{5{x^2} - 10xy}}{{{x^2} - 2xy + 4{y^2}}}.\)

b) \(M = \frac{-1}{{2(x - 1)}}.\)

c) \(M = \frac{{x -y}}{{x + 2y}}.\)

C a) \(M = 5x(x-2y).\)

b) \(M = \frac{1}{{2(x +1)}}.\)

c) \(M = \frac{{x - y}}{{x - 2y}}.\)

D a) \(M = 5x(x-2y).\)

b) \(M = \frac{1}{{2(x - 1)}}.\)

c) \(M = \frac{{x + y}}{{x + 2y}}.\)

Câu 5 : Rút gọn biểu thức sau:\(Q = 1 + \left( {\frac{{x + 1}}{{{x^3} + 1}} - \frac{1}{{x - {x^2} - 1}} - \frac{2}{{x + 1}}} \right):\frac{{{x^3} - 2{x^2}}}{{{x^3} - {x^2} + x}}\)

A \(Q = \frac{2{x - 1}}{{x + 1}}.\)

B \(Q = \frac{{2x - 1}}{{x + 1}}.\)

C \(Q = \frac{{x - 1}}{{x + 1}}\).

D \(Q = \frac{2x}{{x + 1}}.\)

Câu 6 : Rút gọn:\(A = \frac{{{5^2} - 1}}{{{3^2} - 1}}:\frac{{{9^2} - 1}}{{{7^2} - 1}}:\frac{{{{13}^2} - 1}}{{{{11}^2} - 1}}:...:\frac{{{{55}^2} - 1}}{{{{53}^2} - 1}}\)

A \(\frac{9}{{28}}\)

B \(\frac{10}{{29}}\)

C \(\frac{3}{{28}}\)

D \(\frac{9}{{2}}\)

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