\(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}}=\sqrt{\frac{1}{a^2}+\frac{2}{ab}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}-\frac{2}{ab}}\)
\(=\sqrt{\left(\frac{1}{a}+\frac{1}{b}\right)^2-2\left(\frac{1}{a}+\frac{1}{b}\right)\left(\frac{1}{a+b}\right)+\frac{1}{\left(a+b\right)^2}+2\left(\frac{1}{a}+\frac{1}{b}\right)\left(\frac{1}{a+b}\right)-\frac{2}{ab}}\)
\(=\sqrt{\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right)^2+2\left(\frac{a+b}{ab}\right)\left(\frac{1}{a+b}\right)-\frac{2}{ab}}\)
\(=\sqrt{\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right)^2+\frac{2}{ab}-\frac{2}{ab}}=\left|\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}\right|\)
P/s: Bài này em đã làm rất kĩ rồi, sai thì chịu nha!