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b) \(\dfrac{\left(n+1\right)!-n!}{\left(n+1\right)!+n!}=\dfrac{n!.\left(n+1\right)-n!}{n!\left(n+1\right)+n!}=\dfrac{n!\left(n+1-1\right)}{n!\left(n+1+1\right)}=\dfrac{n}{n+2}\)
a) \(\dfrac{8a^{n+2}+a^{n-1}}{16a^{n+4}+4a^{n+2}+a^n}=\dfrac{8a^{n-1+3}+a^{n-1}}{16a^{n-1+5}+4a^{n-1+3}+a^{n-1+1}}\)
\(=\dfrac{8a^{n-1}.a^3+a^{n-1}}{16a^{n-1}a^5+4a^{n-1}a^3+a^{n-1}a}=\dfrac{a^{n-1}\left(8a^3+1\right)}{a^{n-1}\left(16a^5+4a^3+a\right)}\)
\(=\dfrac{8a^3+1}{16a^5+4a^3+a}\)
a) \(a^4-5a^2+4=\)\(\left(a^4-4a^2\right)-\left(a^2-4\right)=a^2\left(a^2-4\right)-\left(a^2-4\right)=\left(a^2-1\right)\left(a^2-4\right)\)
\(=\left(a-1\right)\left(a+1\right)\left(a-2\right)\left(a+2\right)\)
\(a^4-a^2+4a-4=a^2\left(a^2-1\right)+4\left(a-1\right)=a^2\left(a-1\right)\left(a+1\right)+4\left(a-1\right)\)
\(=\left(a-1\right)\left[a^2\left(a+1\right)+4\right]=\left(a-1\right)\left(a^3+a^2+4\right)\)
\(a^3+a^2+4=\left(a^3+2a^2\right)-\left(a^2+2a\right)+\left(2a+4\right)=a^2\left(a+2\right)-a\left(a+2\right)+2\left(a+2\right)\)
\(=\left(a^2-a+2\right)\left(a+2\right)\)
\(N=\frac{\left(a-1\right)\left(a+1\right)\left(a-2\right)\left(a+2\right)}{\left(a-1\right)\left(a+2\right)\left(a^2-a+2\right)}=\frac{\left(a+1\right)\left(a-2\right)}{a^2-a+2}\)
bài 4: Ta có \(x^2-2y^2=xy\Rightarrow x^2-y^2=xy+y^2\Rightarrow\left(x-y\right)\left(x+y\right)=y\left(x+y\right)\)
\(x-y=y\Rightarrow x=2y\)
thay x=2y vào A ta đc :
A = \(\dfrac{x-y}{x+y}=\dfrac{2y-y}{2y+y}=\dfrac{y}{3y}=\dfrac{1}{3}\)
Bài 1:
Ta có: \(x+y+z=0\Rightarrow z=-x-y\Rightarrow z^2=(-x-y)^2\)
\(\Rightarrow x^2+y^2-z^2=x^2+y^2=x^2+y^2-(-x-y)^2=-2xy\)
Hoàn toàn tương tự:
\(y^2+z^2-x^2=-2yz; z^2+x^2-y^2=-2xz\)
Do đó:
\(P=\frac{(x^2+y^2-z^2)(y^2+z^2-x^2)(z^2+x^2-y^2)}{16xyz}=\frac{(-2xy)(-2yz)(-2xz)}{16xyz}=\frac{-xyz}{2}\)
c)\(P=\)\(\frac{\left(a-b\right)^2-c^2}{\left(a-b+c\right)^2}=\frac{\left(a-b+c\right)\left(a-b-c\right)}{\left(a-b+c\right)^2}=\frac{a-b-c}{a-b+c}\)
b)\(M\)\(=\frac{\left(a+2\right)\left(a-1\right)^2}{\left(2a-3\right)\left(a-1\right)^2}=\frac{a+2}{2a-3}\)
\(\frac{1^2}{2^2-1}\cdot\frac{3^2}{4^2-1}\cdot\cdot\cdot\cdot\cdot\frac{n^2}{\left(n+1\right)^2-1}\)
\(=\frac{1\cdot1}{1\cdot3}\cdot\frac{3\cdot3}{3\cdot5}\cdot\cdot\cdot\cdot\cdot\frac{n\cdot n}{n\left(n+2\right)}\)
\(=\frac{\left(1\cdot3\cdot\cdot\cdot\cdot\cdot n\right)\left(1\cdot3\cdot\cdot\cdot\cdot\cdot n\right)}{\left(1\cdot3\cdot\cdot\cdot\cdot\cdot n\right)[3\cdot5\cdot\cdot\cdot\cdot\cdot(n+2)]}\)
\(=\frac{1}{n+2}\)