a) \(\frac{6}{x^2+4x}+\frac{3}{2x+8}=\frac{6.2}{2x\left(x+4\right)}+\frac{3x}{2x\left(x+4\right)}=\frac{12+3x}{2x\left(x+4\right)}=\frac{3\left(x+4\right)}{2x\left(x+4\right)}=\frac{3}{2x}\)

c) \(\frac{-5}{4+2y}+\frac{y-2}{2y+y^2}=\frac{-5.y}{2y\left(y+2\right)}+\frac{2\left(y-2\right)}{2y\left(y+2\right)}=\frac{-5y+2y-4}{2y\left(y+2\right)}=\frac{-3y-4}{2y\left(y+2\right)}\)

d) \(\frac{x-1}{x^2-2xy}+\frac{3}{2xy-x^2}=\frac{x-1}{x\left(x-2y\right)}-\frac{3}{x\left(x-2y\right)}=\frac{x-1-3}{x\left(x-2y\right)}=\frac{x-4}{x\left(x-2y\right)}\)

\(4x^2-20x+25-4x^2-x+8x+2=0\)

\(-13x+27=0\)

\(x=\frac{27}{13}\)

\(4x^2-12x+9-4x^2+15=0\)

\(-12x+24=0\)

\(-12\left(x-2\right)=0\)

\(\Rightarrow x-2=0\)

\(x=2\)

bấm máy tính là ra bn

2 nhé

hok tốt

nhớ t.i.c.k

b) rút gọn còn :

\(\frac{10xy^5}{12xy}\)

\(a.=\frac{2x}{5y}\)

\(b.=\frac{5y^4}{6}\)

\(c.=\frac{\left(x+1\right)^2}{5x^2\left(x+1\right)}=\frac{x+1}{5x^2}\)

\(d.=\frac{2x\left(x+1\right)}{x+1}=2x\)

\(e.=\frac{-3\left(x-y\right)}{\left(x-y\right)}=-3\)

\(f.=\frac{-x\left(x-1\right)}{\left(x-1\right)}=-x\)

Ta có:

M = \(\frac{1}{1-x}\cdot\frac{1}{1+x}\cdot\frac{1}{1+x^2}\cdot\frac{1}{1+x^4}\cdot\frac{1}{1+x^8}\cdot\frac{1}{1+x^{16}}\)

M = \(\frac{1}{\left(1-x\right)\left(1+x\right)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)}\)

M = \(\frac{1}{\left(1-x^2\right)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)}\)

M = \(\frac{1}{\left(1-x^4\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)}\)

M = \(\frac{1}{\left(1-x^8\right)\left(1+x^8\right)\left(1+x^{16}\right)}\)

M = \(\frac{1}{\left(1-x^{16}\right)\left(1+x^{16}\right)}\)

M = \(\frac{1}{1-x^{32}}\)

chỉ cần CM \(Q=2^{2^n}+4^n+1⋮3\) là ok 

Với n=1 thì \(Q⋮3\)

Giả sử Q vẫn chia hết cho 3 đến n=k, ta có: \(Q=2^{2^k}+4^k+1⋮3\)

Với n=k+1 thì \(Q=2^{2^k.2}+4^{k+1}+1=2^{2^k}.2^{2^k}+4^k.4+1\)

\(=\left(2^{2^k}.2^{2^k}+2^{2^k}.4^k+2^{2^k}\right)-\left(2^{2^k}.4^k+2^{2^k}-4^k.4-4\right)-3\)

\(=2^{2^k}\left(2^{2^k}+4^k+1\right)-\left(4^k+1\right)\left(2^{2^k}-4\right)-3\)

\(=2^{2^k}Q-\left(4^k+1\right)\left(4^{2^{k-1}}-1-3\right)-3⋮3\) do \(\left(4^{2^{k-1}}-1\right)⋮\left(4-1\right)=3\)

b)Ta có:  \(a^{2000}+b^{2000}=a^{2001}+b^{2001}\)

\(\Rightarrow a^{2001}+b^{2001}\)\(-a^{2000}-b^{2000}=0\)

\(\Rightarrow a^{2000}\left(a-1\right)+b^{2000}\left(b-1\right)=0\)(1)

và \(a^{2001}+b^{2001}=a^{2002}+b^{2002}\)

\(\Rightarrow a^{2002}+b^{2002}\)\(-a^{2001}-b^{2001}=0\)

\(\Rightarrow a^{2001}\left(a-1\right)+b^{2001}\left(b-1\right)=0\)(2)

Lấy (2) - (1), ta được: \(a^{2000}\left(a-1\right)^2+b^{2000}\left(b-1\right)^2=0\)(3)

Mà \(a^{2000}\left(a-1\right)^2\ge0\forall a\)và \(b^{2000}\left(b-1\right)^2\ge0\forall b\)

nên (3) xảy ra\(\Leftrightarrow\hept{\begin{cases}a^{2000}\left(a-1\right)^2=0\\b^{2000}\left(b-1\right)^2=0\end{cases}}\Leftrightarrow\hept{\begin{cases}a=1hoaca=0\\b=1hoacb=0\end{cases}}\)

Mà a,b dương nên a = 1 và b = 1

a) Áp dụng BĐT Svac - xơ:

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{\left(1+1+1\right)^2}{a+b+c}=9\)

(Dấu "="\(\Leftrightarrow a=b=c=\frac{1}{3}\))