Hãy rút gọn biểu thức:
a) \(\left(3x^{n+1}-y^{n-1}\right)-3\left(x^{n+1}+5y^{n-1}\right)+4\left(x^{n+1}+2y^{n-1}\right)\)
b) \(\left(\dfrac{3}{4}x^{n+1}-\dfrac{1}{2}y^n\right)\times2xy-\left(\dfrac{2}{3}x^{n+1}-\dfrac{5}{6}y^n\right)\times7xy\)
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Những câu hỏi liên quan
NV
Nguyễn Việt Lâm
Giáo viên
22 tháng 4 2022
\(S\left(x\right)=\dfrac{1}{x^2}+\dfrac{2}{x^3}+...+\dfrac{n}{x^{n+1}}\)
\(\Rightarrow x.S\left(x\right)=\dfrac{1}{x}+\dfrac{2}{x^2}+\dfrac{3}{x^3}+...+\dfrac{n}{x^n}\)
\(\Rightarrow x.S\left(x\right)-S\left(x\right)=\dfrac{1}{x}+\dfrac{1}{x^2}+\dfrac{1}{x^3}+...+\dfrac{1}{x^n}-\dfrac{n}{x^{n+1}}\)
\(\Rightarrow\left(x-1\right)S\left(x\right)=\dfrac{1}{x}.\dfrac{1-\left(\dfrac{1}{x}\right)^n}{1-\dfrac{1}{x}}-\dfrac{n}{x^{n+1}}=\dfrac{x^n-1}{x^n\left(x-1\right)}-\dfrac{n}{x^{n+1}}=\dfrac{x^{n+1}-x-n\left(x-1\right)}{x^{n+1}\left(x-1\right)}\)
\(\Rightarrow S\left(x\right)=\dfrac{x^{n+1}-\left(n+1\right)x+n}{x^{n+1}\left(x-1\right)^2}\)
HV
20 tháng 11 2017
1/
\(\dfrac{\left(x-y\right)^3-3xy\left(x+y\right)+y^3}{x-6y}\)
\(=\dfrac{x^3-3x^2y+3xy^2-y^3-3x^2y-3xy^2+y^3}{x-6y}\)
\(=\dfrac{x^3-6x^2y}{x-6y}\)
\(=\dfrac{x^2\left(x-6y\right)}{x-6y}\)
\(=x^2\)
\(2\)/
\(\dfrac{x^2+y^2+z^2-2xy+2xz-2yz}{x^2-2xy+y^2-z^2}\)
\(=\dfrac{\left(x-y+z^{ }\right)^2}{\left(x-y\right)^2-z^2}\)
\(=\dfrac{\left(x-y+z\right)^2}{\left(x-y-z\right)\left(x-y+z\right)}\)
\(=\dfrac{x-y+z}{x-y-z}\)
3/
\(\dfrac{\left(n+1\right)!}{n!\left(n+2\right)}\)
\(=\dfrac{n!\left(n+1\right)}{n!\left(n+2\right)}\)
\(=\dfrac{n+1}{n+2}\)
4/
\(\dfrac{n!}{\left(n+1\right)!-n!}\)
\(=\dfrac{n!}{n!\left(n+1\right)-n!}\)
\(=\dfrac{n!}{n!\left[\left(n+1\right)-1\right]}\)
\(=\dfrac{n!}{n!.n}\)
\(=\dfrac{1}{n}\)
5/
\(\dfrac{\left(n+1\right)!-\left(n+2\right)!}{\left(n+1\right)!+\left(n+2\right)!}\)
\(=\dfrac{\left(n+1\right)!-\left(n+1\right)!\left(n+2\right)}{\left(n+1\right)!+\left(n+1\right)!\left(n+2\right)}\)
\(=\dfrac{\left(n+1\right)!\left(-n-1\right)}{\left(n+1\right)!\left(n+3\right)}\)
\(=\dfrac{-n-1}{n+3}\)
25 tháng 7 2017
143. a) \(-6x^n.y^n.\left(-\dfrac{1}{18}x^{2-n}+\dfrac{1}{72}y^{5-n}\right)\)
\(=-6.\left(-\dfrac{1}{18}\right)x^n.x^{2-n}.y^n+\left(-6\right).\dfrac{1}{27}x^n.y^n.y^{5-n}\)
\(=\dfrac{1}{3}x^{n+2-n}y^n-\dfrac{2}{9}x^n.y^{n+5-n}\)
\(=\dfrac{1}{3}x^2y^n-\dfrac{2}{9}x^ny^5\)
b) Ta có: \(\left(5x^2-2y^2-2xy\right)\left(-xy-x^2+7y^2\right)\)
\(=5x^2\left(-xy\right)+5x^2.\left(-x^2\right)+5x^2.7y^2-2y^2.\left(-xy\right)-2y^2.\left(-x^2\right)-2y^2.7y^2-2xy.\left(-xy\right)-2xy\left(-x^2\right)-2xy.7y^2\)
\(=-5x^3y-5x^4+35x^2y^2+2xy^3+2x^2y^2-14y^4+2x^2y^2+2x^3y-14xy^3\)
Rút gọn các đa thức đồng dạng, ta có kết quả:
\(-5x^4-3x^3y+39x^2y^2-12xy^3-14y^4\)
Kết quả đã được xếp theo lũy thừa giảm dần của x
PN
12 tháng 8 2017
+ Biểu thức nguyên:
\(3x^2y-5x\left(y^2+3\right)\)
\(\dfrac{1}{3a}x^n-4x^2+\dfrac{x}{a+3}\)
+ Biểu thức phân:
\(\dfrac{5\left(x-4\right)\left(x^2+y\right)}{6xy-1}\)
TN
23 tháng 7 2017
\(A=\dfrac{x^2}{\left(x+y\right)\left(1-y\right)}-\dfrac{y^2}{\left(x+y\right)\left(1+x\right)}-\dfrac{x^2y^2}{\left(1+x\right)\left(1-y\right)}\)đkxđ: \(y\ne1;x\ne-1;x\ne-y\)\(=\dfrac{x^2\left(1+x\right)}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}-\dfrac{y^2\left(1-y\right)}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}-\dfrac{x^2y^2\left(x+y\right)}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}\)\(=\dfrac{x^2+x^3-y^2+y^3-x^3y^2-x^2y^3}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}\)
\(=\dfrac{\left(x^3+y^3\right)+\left(x^2-y^2\right)-\left(x^3y^2+x^2y^3\right)}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}\)
\(=\dfrac{\left(x+y\right)\left(x^2-xy+y^2\right)+\left(x-y\right)\left(x+y\right)-x^2y^2\left(x+y\right)}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}\)\(=\dfrac{\left(x+y\right)\left(x^2-xy+y^2+x-y-x^2y^2\right)}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}\)\(=\dfrac{\left(x^2+x\right)-\left(xy+y\right)+\left(y^2-x^2y^2\right)}{\left(1-y\right)\left(x+1\right)}\)
\(=\dfrac{x\left(x+1\right)-y\left(x+1\right)-y^2\left(x-1\right)\left(x+1\right)}{\left(1-y\right)\left(x+1\right)}\) \(=\dfrac{\left(x+1\right)\left(x-y-y^2x+y^2\right)}{\left(1-y\right)\left(x+1\right)}\)
\(=\dfrac{-\left(y-y^2\right)+\left(x-y^2x\right)}{1-y}\)
\(=\dfrac{-y\left(1-y\right)+x\left(1-y\right)\left(1+y\right)}{1-y}\)
\(=\dfrac{\left(1-y\right)\left(x+xy-y\right)}{1-y}=x+xy-y\)
9 tháng 10 2017
\(\dfrac{x^2}{\left(x+y\right)\left(1-y\right)}-\dfrac{y^2}{\left(x+y\right)\left(1+x\right)}-\dfrac{x^2y^2}{\left(1+x\right)\left(1-y\right)}\) MTC : (x+y)(1-y)(1+x)
A=
\(\dfrac{x^2\times\left(1+x\right)}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}-\dfrac{y^2\times\left(1-y\right)}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}-\dfrac{x^2y^2\times\left(x+y\right)}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}\)
A= \(\dfrac{x^2+x^3}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}-\dfrac{y^2}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}-\dfrac{x^3y^2+x^2y^3}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}\)
\(\dfrac{x^2+x^3-y^2-x^3y^2-x^2y^3}{\left(x+y\right)\left(1-y\right)\left(1+x\right)}\)
a) \(\left(3x^{n+1}-y^{n-1}\right)-3\left(x^{n+1}+5y^{n-1}\right)-4\left(x^{n+1}+2y^{n-1}\right)\)
\(=3x^{n+1}-y^{n-1}-3x^{n+1}-15y^{n-1}+4x^{n+1}+8y^{n-1}\)
\(=-8y^{n-1}+4x^{n+1}\)
b) \(\left(\dfrac{3}{4}x^{n+1}-\dfrac{1}{2}y^n\right)\cdot2xy-\left(\dfrac{2}{3}x^{n+1}-\dfrac{5}{6}y^n\right)\cdot7xy\)
\(=\dfrac{3}{2}x^{n+2}y-xy^{n+1}+\left(-\dfrac{2}{3}x^{n+1}-\dfrac{5}{6}y^n\right)\cdot7xy\)
\(=\dfrac{3}{2}x^{n+2}y-xy^{n+1}-\dfrac{14}{3}x^{n+2}y+\dfrac{35}{6}xy^{n+1}\)
\(=-\dfrac{19}{6}x^{n+2}y+\dfrac{29}{6}xy^{n+1}\)
a)\(\left(3x^{n+1}-y^{n-1}\right)-3\left(x^{n+1}+5y^{n-1}\right)+4\left(x^{n+1}+2y^{n-1}\right)\)
\(=3x^{n+1}-y^{n-1}-3x^{n+1}-15y^{n-1}+4x^{n+1}+8y^{n-1}\)
\(=4x^{n+1}-8y^{n-1}\) \(\left(=4\left(x^{n+1}-2y^{n-1}\right)\right)\)