Thực hiện phép cộng cac phân thức sau:
a) 5 y + 3 y 2 − 3 y + 9 − y 9 − 3 y với y ≠ 0 và y ≠ 3
b) 6 m 2 + 11 m + 4 m 3 − 1 + 2 m − 1 m 2 + m + 1 + 7 1 − m với m ≠ 1 .
Thực hiện các phép tính cộng, trừ phân thức sau:
a) \(\dfrac{x}{{x + 3}} + \dfrac{{2 - x}}{{x + 3}}\) b) \(\dfrac{{{x^2}y}}{{x - y}} - \dfrac{{x{y^2}}}{{x - y}}\) c) \(\dfrac{{2x}}{{2x - y}} + \dfrac{y}{{y - 2x}}\)
\(a,\dfrac{x}{x+3}+\dfrac{2-x}{x+3}\\ =\dfrac{x+2-x}{x+3}\\ =\dfrac{2}{x+3}\\b,\dfrac{x^2y}{x-y}-\dfrac{xy^2}{x-y}\\ =\dfrac{x^2y-xy^2}{x-y}\\ =\dfrac{xy\left(x-y\right)}{x-y}\\ =xy\\ c,\dfrac{2x}{2x-y}+\dfrac{y}{y-2x}\\=\dfrac{2x}{2x-y}-\dfrac{y}{2x-y}\\ =\dfrac{2x-y}{2x-y}\\ =1 \)
`a, x/(x+3) + (2-x)/(x+3) = (x+2-x)/(x+3) = 2/(x+3)`
`b, (x^2y)/(x-y) - (xy^2)/(x-y) = (x^2y-xy^2)/(x-y) = (xy(x-y))/(x-y)= xy`
`c, (2x)/(2x-y) - (y)/(2x-y)`
`= (2x-y)/(2x-y) = 1`
Thực hiện các phép cộng, trừ phân thức sau:
a) \(\dfrac{1}{{2a}} + \dfrac{2}{{3b}}\)
b) \(\dfrac{{x - 1}}{{x + 1}} - \dfrac{{x + 1}}{{x - 1}}\)
c) \(\dfrac{{x + y}}{{xy}} - \dfrac{{y + z}}{{yz}}\)
d) \(\dfrac{2}{{x - 3}} - \dfrac{{12}}{{{x^2} - 9}}\)
e) \(\dfrac{1}{{x - 2}} + \dfrac{2}{{{x^2} - 4x + 4}}\)
a: \(=\dfrac{3b+4a}{6ab}\)
b: \(=\dfrac{x^2-2x+1-x^2-2x-1}{x^2-1}=\dfrac{-4x}{x^2-1}\)
c: \(=\dfrac{xz+yz-xy-xz}{xyz}=\dfrac{yz-xy}{xyz}=\dfrac{z-x}{xz}\)
d: \(=\dfrac{2x+6-12}{\left(x-3\right)\left(x+3\right)}=\dfrac{2x-6}{\left(x-3\right)\left(x+3\right)}=\dfrac{2}{x+3}\)
e: \(=\dfrac{x-2+2}{\left(x-2\right)^2}=\dfrac{x}{\left(x-2\right)^2}\)
Thực hiện các phép nhân phân thức sau:
a) \(\dfrac{{4y}}{{3{x^2}}} \cdot \dfrac{{5{x^3}}}{{2{y^3}}}\)
b) \(\dfrac{{{x^2} - 2x + 1}}{{{x^2} - 1}} \cdot \dfrac{{{x^2} + x}}{{x - 1}}\)
c) \(\dfrac{{2x + {x^2}}}{{{x^2} - x + 1}} \cdot \dfrac{{3{x^3} + 3}}{{3x + 6}}\)
\(a,=\dfrac{4y.5x^3}{3x^2.2y^3}=\dfrac{20x^3y}{6x^2y^3}=\dfrac{10x}{3y^2}\\ b,=\dfrac{\left(x-1\right)^2.x\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x-1\right)}=\dfrac{\left(x-1\right)^2.x.\left(x+1\right)}{\left(x-1\right)^2.\left(x+1\right)}=x\)
\(c,=\dfrac{x\left(2+x\right).3\left(x^3+1\right)}{\left(x^2-x+1\right).3.\left(x+2\right)}=\dfrac{3x.\left(x+2\right).\left(x+1\right)\left(x^2-x+1\right)}{\left(x^2-x+1\right).3\left(x+2\right)}=x\left(x+1\right)\)
Thực hiện các phép cộng, trừ phân thức sau:
a) \(\dfrac{a}{{a - 3}} - \dfrac{3}{{a + 3}}\) b) \(\dfrac{1}{{2x}} + \dfrac{2}{{{x^2}}}\) c) \(\dfrac{4}{{{x^2} - 1}} - \dfrac{2}{{{x^2} + x}}\)
`a, a/(a-3) - 3/(a+3) = (a(a+3) - 3(a-3))/(a^2-9)`
`= (a^2+9)/(a^2-9)`
`b, 1/(2x) + 2/x^2 = x/(2x^2) + 4/(2x^2) = (x+4)/(2x^2)`
`c, 4/(x^2-1) - 2/(x^2+x) = (4x)/(x(x-1)(x+1)) - (2(x-1))/(x(x+1)(x-1))`
`= (2x+2)/(x(x-1)(x+1)`
`= 2/(x(x-1))`
Thực hiện các phép chia phân thức sau:
a) \(\dfrac{{5x}}{{4{y^3}}}:\left( { - \dfrac{{{x^4}}}{{20y}}} \right)\)
b) \(\dfrac{{{x^2} - 16}}{{x + 4}} :\dfrac{{2x - 8}}{x}\)
c) \(\dfrac{{2x + 6}}{{{x^3} - 8}}:\dfrac{{{{\left( {x + 3} \right)}^3}}}{{2x - 4}}\)
\(a,=\dfrac{5x}{4y^3}\times\left(\dfrac{-20y}{x^4}\right)=\dfrac{-100xy}{4x^4y^3}=\dfrac{-25}{x^3y^2}\\ b,=\dfrac{\left(x-4\right)\left(x+4\right)}{\left(x+4\right)}\times\dfrac{x}{2\left(x-4\right)}=\dfrac{x}{2}\)
\(c,=\dfrac{2\left(x+3\right)}{\left(x-2\right)\left(x^2+2x+4\right)}\times\dfrac{2\left(x-2\right)}{\left(x+3\right)^3}=\dfrac{4}{\left(x+3\right)^2.\left(x^2+2x+4\right)}\)
a) \(\dfrac{5x}{4y^3}:\left(-\dfrac{x^4}{20y}\right)=\dfrac{5x}{4y^3}\cdot\left(-\dfrac{20y}{x^4}\right)=\dfrac{5\cdot-5}{y^2\cdot x^3}=\dfrac{-25}{x^3y^2}\)
b) \(\dfrac{x^2-16}{x+4}:\dfrac{2x-8}{x}=\left(x-4\right)\cdot\dfrac{x}{2\left(x-4\right)}=\dfrac{x}{2}\)
c) \(\dfrac{2x+6}{x^3-8}:\dfrac{\left(x+3\right)^3}{2x-4}=\dfrac{2\left(x+3\right)}{\left(x-2\right)\left(x^2+2x+4\right)}\cdot\dfrac{2\left(x-2\right)}{\left(x+3\right)^3}=\dfrac{4}{\left(x^2+2x+4\right)\left(x+3\right)^2}\)
Thực hiện các phép cộng, trừ phân thức sau:
a) \(\dfrac{{a - 1}}{{a + 1}} + \dfrac{{3 - a}}{{a + 1}}\) b) \(\dfrac{b}{{a - b}} + \dfrac{a}{{b - a}}\) c) \(\dfrac{{{{\left( {a + b} \right)}^2}}}{{ab}} - \dfrac{{{{\left( {a - b} \right)}^2}}}{{ab}}\)
a) \(\dfrac{a-1}{a+1}+\dfrac{3-a}{a+1}\)
\(=\dfrac{a-1+3-a}{a+1}\)
\(=\dfrac{2}{a+1}\)
b) \(\dfrac{b}{a-b}+\dfrac{a}{b-a}\)
\(=\dfrac{b}{a-b}+\dfrac{-a}{a-b}\)
\(=\dfrac{b-a}{a-b}\)
\(=-1\)
c) \(\dfrac{\left(a+b\right)^2}{ab}-\dfrac{\left(a-b\right)^2}{ab}\)
\(=\dfrac{\left[\left(a+b\right)-\left(a-b\right)\right]\left[\left(a+b\right)+\left(a-b\right)\right]}{ab}\)
\(=\dfrac{4ab}{ab}\)
\(=4\)
`a, (a-1)/(a+1) + (3-a)/(a+1)`
`= (a-1+3-a)/(a+1)`
`=2/(a+1)`
`b, b/(a-b) + a/(b-a)`
`= b/(a-b) - a/(a-b)`
`= (b-a)/(a-b)`
`c, (a+b)^2/(ab) -(a-b)^2/(ab)`
`=(a^2+2ab+b^2-a^2+2ab-b^2)/(ab)`
`= (4ab)/(ab)`
Thực hiện các phép nhân đơn thức sau:
a) \(\left( {4{x^3}} \right).\left( { - 6{x^3}y} \right)\) b) \(\left( { - 2y} \right).\left( { - 5x{y^2}} \right)\) c) \({\left( { - 2a} \right)^3}.{\left( {2ab} \right)^2}\)
`a)`
`4x^3 * (-6x^3y)`
`= 4*(-6) * (x^3*x^3) * y`
`= -24x^6y`
`b)`
`(-2y)*(-5xy^2)`
`= (-2)*(-5)*x*(y*y^2)`
`= 10xy^3`
`c)`
`(-2a)^3 * (2ab)^2`
`= (-8a^3) * (4a^2b^2)`
`= (-8*4)*(a^3*a^2)*b^2`
`= -32a^5b^2`
a) \(4x^3\cdot\left(-6x^3y\right)\)
\(=\left(4\cdot-6\right)\cdot\left(x^3\cdot x^3\right)\cdot y\)
\(=-24x^6y\)
b) \(\left(-2y\right)\cdot\left(-5xy^2\right)\)
\(=\left(-2\cdot-5\right)\cdot\left(y\cdot y^2\right)\cdot x\)
\(=10xy^3\)
c) \(\left(-2a\right)^3\cdot\left(2ab\right)^2\)
\(=-8a^3\cdot4a^2b^2\)
\(=\left(-8\cdot4\right)\cdot\left(a^3\cdot a^2\right)\cdot b^2\)
\(=-32a^5b^2\)
Thực hiện phép cộng các phân thức sau:
a) u + 10 u − 2 + u − 18 u − 2 + u + 2 u 2 − 4 với u ≠ ± 1 2 ;
b) 2 − x 2 x 2 y 2 + 5 + 2 y 8 x 3 y 2 + x − 7 4 x 3 y với x ≠ 0 và y ≠ 0 .
Thực hiện mỗi phép tính sau:
a) \({x^2} + \dfrac{1}{4}{x^2} - 5{x^2}\);
b) \({y^4} + 6{y^4} - \dfrac{2}{5}{y^4}\).
a) \({x^2} + \dfrac{1}{4}{x^2} - 5{x^2} = (1 + \dfrac{1}{4} - 5){x^2} = - \dfrac{{15}}{4}{x^2}\);
b) \({y^4} + 6{y^4} - \dfrac{2}{5}{y^4} = (1 + 6 - \dfrac{2}{5}){y^4} = \dfrac{{33}}{5}{y^4}\).