\(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}=\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2\left(a+b+c\right)}{abc}}\) ( do \(a+b+c=0\) )
\(=\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2}{ab}+\frac{2}{bc}+\frac{2}{ca}}\)
\(=\sqrt{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2}\)
\(=\left|\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right|\) ( đpcm )
Cho a+b+c=0; a,b,c≠0. Chứng minh \(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}=\left|\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right|\)
cho a,b,c>0 và \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le16\left(a+b+c\right)\). Chứng minh rằng:
\(\frac{1}{\left(a+b+2\sqrt{a+c}\right)^3}+\frac{1}{\left(b+c+2\sqrt{b+a}\right)^3}+\frac{1}{\left(c+a+2\sqrt{b+c}\right)^3}\le\frac{8}{9}\)
Cho a, b, c > 0 thoả mãn: a + b + c = \(\sqrt{a}+\sqrt{b}+\sqrt{c}\) = 2. Chứng minh: \(\frac{\sqrt{a}}{a+1}+\frac{\sqrt{b}}{b+1}+\frac{\sqrt{c}}{c+1}=\frac{2}{\sqrt{\left(1+a\right)\left(1+b\left(1+c\right)\right)}}\)
Cho a, b, c > 0 thoả mãn a + b + c = \(\sqrt{a}+\sqrt{b}+\sqrt{c}\) = 2. Chứng minh: \(\frac{\sqrt{a}}{a+1}+\frac{\sqrt{b}}{b+1}+\frac{\sqrt{c}}{c+1}=\frac{2}{\sqrt{\left(a+1\right)\left(b+1\right)\left(c+1\right)}}\)
Cho a,b,c>0 thỏa mãn: a.b.c=8
Chứng minh: \(\frac{a^2}{\sqrt{\left(1+a^3\right).\left(1+b^3\right)}}+\frac{b^2}{\sqrt{\left(1+b^3\right).\left(1+c^3\right)}}+\frac{c^2}{\sqrt{\left(1+c^3\right).\left(1+a^3\right)}}\ge\frac{4}{3}\)
Cho a+b+c=0 và \(a,b,c\ne0\) . Chứng minh đẳng thức
\(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}=\left|\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right|\)
Chứng minh: \(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{\left(a+b+c\sqrt[3]{abc}\right)^2}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\) với mọi a,b,c >0
1, cho a,b là số thực thỏa 0<a<1 ; 0<b<1 ; a khác b ; và \(a-b=\sqrt{1-b^2}-\sqrt{1-a^2}\)
tính giá trị biểu thức Q= \(\sqrt{a^2+b^2}+2019\)
2 Tìm nghiệm nguyen pt \(x^2-4xy+5y^2=2\left(x-y\right)\)
3. cho a,b,c>0 chứng minh \(\frac{bc}{a^2\left(b+c\right)}+\frac{ca}{b^2\left(c+a\right)}+\frac{ab}{c^2\left(a+b\right)}\ge\frac{1}{2a}+\frac{1}{2b}+\frac{1}{2c}\)
C/Minh đẳng thức:
a) \(\left(\frac{\sqrt{a}+2}{a+2\sqrt{a}+1}-\frac{\sqrt{a}-2}{a-1}\right).\frac{\sqrt{a}+1}{\sqrt{a}}=\frac{2}{a-1}\) (với a>0, b>0, a≠b)
b)\(\frac{2}{\sqrt{ab}}:\left(\frac{1}{\sqrt{a}}-\frac{1}{\sqrt{b}}\right)^2-\frac{a+b}{\left(\sqrt{a}-\sqrt{b}\right)^2}=-1\) (với a>0, b>0,a≠b)
c) \(\frac{2\sqrt{a}+3\sqrt{b}}{\sqrt{ab}+2\sqrt{a}-3\sqrt{b}-6}-\frac{6-\sqrt{ab}}{\sqrt{ab}+2\sqrt{a}+3\sqrt{b}+6}=\frac{a+9}{a-9}\) (với a≥0, b≥0,a≠9)
