Có: \(\left(a-b\right)^2\ge0,\forall a,b\)
\(\Rightarrow a^2+b^2\ge2ab\)
\(\Rightarrow2\left(a^2+b^2\right)\ge a^2+b^2+2ab\)
\(\Rightarrow2\left(a^2+b^2\right)\ge\left(a+b\right)^2\)
\(\Rightarrow\sqrt{2\left(a^2+b^2\right)}\ge a+b\)
\(\Leftrightarrow\sqrt{a^2+b^2}\ge\frac{a+b}{\sqrt{2}}\)
Bài 1 : Cho 3 số dương a,b,c thỏa mãn : \(b\ne c;\sqrt{a}+\sqrt{b}\ne\sqrt{c}\) và \(a+b=\left(\sqrt{a}+\sqrt{b}-\sqrt{c}\right)^2\)
CMR : \(\frac{a+\left(\sqrt{a}-\sqrt{c}\right)^2}{b+\left(\sqrt{b}-\sqrt{c}\right)^2}=\frac{\sqrt{a}-\sqrt{c}}{\sqrt{b}-\sqrt{c}}\)
Cho 3 số thực dương a,b,c thỏa mãn : \(a+b+c=\sqrt{a}+\sqrt{b}+\sqrt{c}=2\) . CMR :
\(\frac{\sqrt{a}}{1+a}+\frac{\sqrt{b}}{1+b}+\frac{\sqrt{c}}{1+c}=\frac{2}{\left(1+a\right)\left(1+b\right)\left(1+c\right)}\)
Cho 3 số thực dương a,b,c thỏa mãn : \(a\sqrt{1-b^2}+b\sqrt{1-c^2}+c\sqrt{1-a^2}=\frac{3}{2}\)
CMR :\(a^2+b^2+c^2=\frac{3}{2}\)
Cho x=\(\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}\) voi a<0 ;b<0
a)CMR:\(x^2-4\ge0\)
b)Rut gon :\(\sqrt{x^2-4}\)
rút gọn P: P=\(\left(\frac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2}-b^2}-\frac{a-\sqrt{a^2-b^2}}{a+\sqrt{a^2-b^2}}\right):\frac{4\sqrt{a^4-a^2b^2}}{b^2}\)
Thu gọn biểu thức
a, A = \(\frac{2\sqrt{3-\sqrt{3+\sqrt{3+\sqrt{48}}}}}{\sqrt{6}-2}\)
b, B = \(\frac{\sqrt{a}+\sqrt{b}-1}{a+\sqrt{ab}}+\frac{\sqrt{a}-\sqrt{b}}{2\sqrt{ab}}\cdot\left(\frac{\sqrt{b}}{a-\sqrt{ab}}+\frac{\sqrt{b}}{a+\sqrt{ab}}\right)\)
CMR: A=B
a) A=\(\sqrt{10+\sqrt{24+\sqrt{40+\sqrt{60}}}}\)
B= \(\sqrt{5}\)+\(\sqrt{3}\)+\(\sqrt{2}\)
b) A=\(\sqrt{3+\sqrt{5}}\)
B= \(\frac{\sqrt{5}+1}{\sqrt{2}}\)
\(\frac{a^3-a--2b-\frac{b^2}{a}}{\left(\frac{1}{\sqrt{a}}-\sqrt{\frac{1}{a}+\frac{b}{a^2}}\right)\left(\sqrt{a}+\sqrt{a+b}\right)}:\left(\frac{a^3+a^2+ab+a^2b}{a^2-b^2}+\frac{b}{a-b}\right)\)
cho \(a\ge0\). CMR:
\(\frac{a^2-\sqrt{a}}{a^2+\sqrt{a}+1}-\frac{a^2+\sqrt{a}}{a^2-\sqrt{a}-1}+a+1=\left(\sqrt{a}-1\right)^2\)
