\(a+b+c=0\Leftrightarrow a+b=-c\) thay vào :
\(a^3+b^3+c\left(a^2+b^2\right)-abc=\left(a+b\right)^3-3ab\left(a+b\right)+c\left[\left(a+b\right)^2-2ab\right]-abc\)
\(=-c^3-3ab.\left(-c\right)+c\left[c^2-2ab\right]-abc\)
\(=-c^3+3abc+c^3-2abc-abc=0\)
1. Cho a,b,c ≠0 thỏa mãn: (a+b+c)2=a2+b2+c2
Rút gọn:
\(M=\dfrac{a^2}{a^2+2bc}+\dfrac{b^2}{b^2+2ca}+\dfrac{c^2}{c^2+2ab}\)
2. Cho a+b+c=0
Rút gọn:
\(A=\dfrac{a^3+b^3+c^3-3abc}{\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3}\)
Rút gọn P = \(\frac{a^3+b^3+c^3}{\left(a-b\right)^2+\left(b-c\right)^2+\left(a-c\right)^2}\)
rút gọn các biểu thức sau:
a) \(\sqrt{\left(2-\sqrt{3}\right)^2}\)
b) \(\sqrt{\left(3-\sqrt{11}\right)^2}\)
c) \(2\sqrt{a^2}\)với a ≥ 0
d) 3\(\sqrt{\left(a-2\right)^2}\)với a < 0
1 . Rút gọn : \(\sqrt{37-20\sqrt{3}}+\sqrt{37+20\sqrt{3}}\)
2 . Cho a , b , c > 0 . CM : \(\frac{ab+bc+ca}{a^2+b^2+c^2}+\frac{\left(a+b+c\right)^2}{abc}\ge28\)
rút gọn bt biết a,b,c dương ; ab=1 và a+b khác 0
\(\frac{1}{\left(a+b\right)^3}.\left(\frac{1}{a^3}+\frac{1}{b^3}\right)+\frac{3}{\left(a+b\right)^4}.\left(\frac{1}{a^2}+\frac{1}{b^2}\right)+\frac{6}{\left(a+b\right)^5}.\left(\frac{1}{a}+\frac{1}{b}\right)\)
Rút gọn biểu thức
a. B = \(\left(\dfrac{a-b}{a^2+ab}-\dfrac{a}{b^2+ab}\right):\left(\dfrac{b^3}{a^3-ab^2}+\dfrac{1}{a+b}\right)\)
b. C = \(a:\left(b-2\right)-\left[\left(a^2+2a+1\right):\left(b^2-4\right)\right].\left[\left(b+2\right):\left(a+1\right)\right]\)
Rút gọn biểu thức
a. B = \(\left(\dfrac{a-b}{a^2+ab}-\dfrac{a}{b^2+ab}\right):\left(\dfrac{b^3}{a^3-ab^2}+\dfrac{1}{a+b}\right)\)
b. C = \(a:\left(b-2\right)-\left[\left(a^2+2a+1\right):\left(b^2-4\right)\right].\left[\left(b+2\right):\left(a+1\right)\right]\)
Rút gọn biểu thức
a. B = \(\left(\dfrac{a-b}{a^2+ab}-\dfrac{a}{b^2+ab}\right):\left(\dfrac{b^3}{a^3-ab^2}+\dfrac{1}{a+b}\right)\)
b. C = \(a:\left(b-2\right)-\left[\left(a^2+2a+1\right):\left(b^2-4\right)\right].\left[\left(b+2\right):\left(a+1\right)\right]\)
1. Cho a,b,c không đồng thời bằng 0 và a+b+c=0. Rút gọn:
\(N=\dfrac{a^2}{a^2-b^2-c^2}+\dfrac{b^2}{b^2-c^2-a^2}+\dfrac{c^2}{c^2-a^2-b^2}\)
2. CMR: Nếu a+b+c=2x thì:
\(\dfrac{1}{x-a}+\dfrac{1}{x-b}+\dfrac{1}{x-c}-\dfrac{1}{x}=\dfrac{abc}{x\left(x-a\right)\left(x-b\right)\left(x-c\right)}\)
