huh

\(a,\frac{13}{x-1}+\frac{5}{2x-2}-\frac{6}{3x-3}=3\)

\(\Leftrightarrow\frac{13}{x-1}+\frac{5}{2\left(x-1\right)}-\frac{6}{3\left(x-1\right)}\)

\(\Leftrightarrow\frac{13.2+5-4}{2\left(x-1\right)}=3\)

\(\Leftrightarrow6\left(x-1\right)=27\)

\(\Leftrightarrow x-1=\frac{9}{2}\Leftrightarrow x=\frac{11}{2}\)

\(b,\frac{2x}{3}-\frac{3}{4}>0\)

\(\Leftrightarrow\frac{8x-9}{12}>0\)

\(\Leftrightarrow8x-9>0\Rightarrow x>\frac{9}{8}\)

Câu 4: 

Thay x=2 và y=-1 vào hệ, ta được:

\(\left\{{}\begin{matrix}2a-b=4\\2b+2=-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}b=-2\\a=1\end{matrix}\right.\)

Để hệ phương trình có nghiệm duy nhất thì \(\dfrac{m}{3}< >-\dfrac{1}{m}\)

=>\(m^2\ne-3\)(luôn đúng)

Ta có: \(\left\{{}\begin{matrix}mx-y=2\\3x+my=3m\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=mx-2\\3x+m\left(mx-2\right)=3m\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=mx-2\\3x+m^2x-2m=3m\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=mx-2\\x\left(m^2+3\right)=5m\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{5m}{m^2+3}\\y=m\cdot\dfrac{5m}{m^2+3}-2\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=\dfrac{5m}{m^2+3}\\y=\dfrac{5m^2-2m^2-6}{m^2+3}=\dfrac{3m^2-6}{m^2+3}\end{matrix}\right.\)

\(\left(x+y\right)\cdot\left(m^2+3\right)+8=0\)

=>\(\dfrac{5m+3m^2-6}{m^2+3}\cdot\left(m^2+3\right)+8=0\)

=>\(3m^2+5m-6+8=0\)

=>\(3m^2+5m+2=0\)

=>(m+1)(3m+2)=0

=>\(\left[{}\begin{matrix}m=-1\\m=-\dfrac{2}{3}\end{matrix}\right.\)

Trong mp đáy, qua B kẻ đường thẳng song song AC, lần lượt cắt DA và DC kéo dài tại E và F

\(\Rightarrow AC||\left(SEF\right)\Rightarrow d\left(AC;SB\right)=d\left(AC;\left(SEF\right)\right)=d\left(A;\left(SEF\right)\right)\)

Gọi I là giao điểm AC và BD

Theo định lý Talet: \(\dfrac{ID}{IB}=\dfrac{DC}{AB}=3\Rightarrow\dfrac{ID}{BD}=\dfrac{3}{4}\)

Cũng theo Talet: \(\dfrac{DA}{DE}=\dfrac{DI}{DB}=\dfrac{3}{4}\Rightarrow AD=\dfrac{3}{4}DE\Rightarrow AE=\dfrac{1}{4}DE\)

\(\Rightarrow d\left(A;\left(SEF\right)\right)=\dfrac{1}{4}d\left(D;\left(SEF\right)\right)\)

Trong tam giác vuông EDF, kẻ \(DH\perp EF\) , trong tam giác vuông SDH, kẻ \(DK\perp SH\)

\(\Rightarrow DK\perp\left(SEF\right)\Rightarrow DK=d\left(D;\left(SEF\right)\right)\)

\(DE=\dfrac{4}{3}AD=\dfrac{4a}{3}\); \(DF=\dfrac{4}{3}DC=4a\)

\(\dfrac{1}{DH^2}=\dfrac{1}{DE^2}+\dfrac{1}{DF^2}=\dfrac{5}{8a^2}\)

\(\dfrac{1}{DK^2}=\dfrac{1}{SD^2}+\dfrac{1}{DH^2}=\dfrac{1}{48a^2}+\dfrac{5}{8a^2}\Rightarrow DK=\dfrac{4a\sqrt{93}}{31}\)

\(\Rightarrow d\left(AC;SB\right)=\dfrac{1}{4}DK=\dfrac{a\sqrt{93}}{31}\)

Bài 3:

a: \(P=x\left(x^2-y\right)+y\left(x-y^2\right)\)

\(=x^3-xy+xy-y^3\)

\(=x^3-y^3\)

Thay \(x=-\frac12;y=-\frac12\) vào P, ta được:

\(P=\left(-\frac12\right)^3-\left(-\frac12\right)^3=\left(-\frac18\right)-\left(-\frac18\right)=-\frac18+\frac18=0\)

b: \(Q=x^2\left(y^3-xy^2\right)+x^2y^2\left(x-y+1\right)\)

\(=x^2y^3-x^3y^2+x^3y^2-x^2y^3+x^2y^2=x^2y^2\)

Thay x=-10; y=-10 vào Q, ta được:

\(Q=\left(-10\right)^2\cdot\left(-10\right)^2=100\cdot100=10000\)

c: \(A=x^3+2xy-2x^3+2y^3+2x^3-y^3\)

\(=\left(x^3-2x^3+2x^3\right)+2xy+\left(2y^3-y^3\right)\)

\(=x^3+2xy+y^3\)

Thay x=2; y=-3 vào A, ta được:

\(A=2^3+2\cdot2\cdot\left(-3\right)+\left(-3\right)^3\)

=8-12-27

=-4-27

=-31

d:

x=1; y=-1

=>\(xy=1\cdot\left(-1\right)=-1\)

\(B=xy+x^2y^2-x^4y^4+x^6y^6-x^8y^8\)

\(=\left(xy\right)+\left(xy\right)^2-\left(xy\right)^4+\left(xy\right)^6-\left(xy\right)^8\)

\(=\left(-1\right)+\left(-1\right)^2-\left(-1\right)^4+\left(-1\right)^6-\left(-1\right)^8\)

=-1+1-1+1-1

=-1

e: x=-1; y=1

=>xy=-1

\(C=xy+x^2y^2+x^3y^3+\cdots+x^{10}y^{10}\)

\(=xy+\left(xy\right)^2+\left(xy\right)^3+\cdots+\left(xy\right)^{10}\)

\(=\left(-1\right)+\left(-1\right)^2+\left(-1\right)^3+\cdots+\left(-1\right)^{10}\)

=-1+1+(-1)+1+...+(-1)+1

=0

f: \(M=2x^2\left(x^2-5\right)+x\left(-2x^3+4x\right)+x^2\left(x+6\right)\)

\(=2x^4-10x^2-2x^4+4x^2+x^3+6x^2\)

\(=x^3\)

Khi x=-4 thì \(M=\left(-4\right)^3=-64\)

g: \(N=x^3\left(y+1\right)-xy\left(x^2-2x+1\right)-x\left(x^2+2xy-3y\right)\)

\(=x^3y+x^3-x^3y+2x^2y-xy-x^3-2x^2y+3xy\)

=2xy

Thay x=8; y=-5 vào N, ta được:

\(N=2\cdot8\cdot\left(-5\right)=-80\)

Bài 1:

d: \(x^2+2xy-3\cdot\left(-xy\right)\)

\(=x^2+2xy+3xy=x^2+5xy\)

e: \(\frac12x^2y\left(2x^3-\frac25xy^2-1\right)\)

\(=\frac12x^2y\cdot2x^3-\frac12x^2y\cdot\frac25xy^2-\frac12x^2y\)

\(=x^5y-\frac15x^3y^3-\frac12x^2y\)

f: \(\left(-xy^2\right)^2\left(x^2-2x+1\right)\)

\(=x^2y^4\left(x^2-2x+1\right)\)

\(=x^2y^4\cdot x^2-x^2y^4\cdot2x+x^2y^4\)

\(=x^4y^4-2x^3y^4+x^2y^4\)

g: (2xy+3)(x-2y)

\(=2xy\cdot x-2xy\cdot2y+3\cdot x-3\cdot2y\)

\(=2x^2y-4xy^2+3x-6y\)

h: \(\left(xy+2y\right)\left(x^2y-2xy+4\right)\)

\(=x^3y^2-2x^2y^2+4xy+2x^2y^2-4xy^2+8y\)

\(=x^3y^2+4xy-4xy^2+8y\)

i: \(4\left(x^2-\frac12y\right)\left(x^2+\frac12y\right)\)

\(=4\left(x^4-\frac14y^2\right)\)

\(=4\cdot x^4-4\cdot\frac14y^2=4x^4-y^2\)

k: \(2x^2\left(1-3x+2x^2\right)\)

\(=2x^2\cdot1-2x^2\cdot3x+2x^2\cdot2x^2\)

\(=2x^2-6x^3+4x^4\)

l: \(\left(2x^2-3x+4\right)\left(-\frac12x\right)\)

\(=-\frac12x\cdot2x^2+3x\cdot\frac12x-4\cdot\frac12x=-x^3+\frac32x^2-2x\)

m: \(\frac12xy\left(-x^3+2xy-4y^2\right)\)

\(=-\frac12xy\cdot x^3+\frac12xy\cdot2xy-\frac12xy\cdot4y^2\)

\(=-\frac12x^4y+x^2y^2-2xy^3\)

n: \(\frac12x^2y\left(2x^3-\frac25xy^2-1\right)\)

\(=\frac12x^2y\cdot2x^3-\frac12x^2y\cdot\frac25xy^2-\frac12x^2y\)

\(=x^5y-\frac15x^3y^3-\frac12x^2y\)