Question

Rút gọn:

K=\(\sqrt{\dfrac{1}{a^2+b^2}+\dfrac{1}{\left(a+b\right)^2}+\sqrt{\dfrac{1}{a^4}+\dfrac{1}{b^4}+\dfrac{1}{\left(a^2+b^2\right)^2}}}\)

Answer 1

\(K=\sqrt{\dfrac{1}{a^2+b^2}+\dfrac{1}{\left(a+b\right)^2}+\sqrt{\dfrac{1}{a^4}+\dfrac{1}{b^4}+\dfrac{1}{\left(a^2+b^2\right)^2}}}\)

\(=\sqrt{\dfrac{1}{a^2+b^2}+\dfrac{1}{\left(a+b\right)^2}+\sqrt{\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}\right)^2-\dfrac{2}{a^2+b^2}\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}\right)+\dfrac{1}{\left(a^2+b^2\right)^2}}}\)

\(=\sqrt{\dfrac{1}{a^2+b^2}+\dfrac{1}{\left(a+b\right)^2}+\sqrt{\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}-\dfrac{1}{a^2+b^2}\right)^2}}\)

\(=\sqrt{\dfrac{1}{\left(a+b\right)^2}+\dfrac{1}{a^2}+\dfrac{1}{b^2}}\)\(=\sqrt{\dfrac{1}{\left(a+b\right)^2}+\left(\dfrac{1}{a}+\dfrac{1}{b}\right)^2-\dfrac{2}{\left(a+b\right)}\left(\dfrac{1}{a}+\dfrac{1}{b}\right)}\)

\(=\sqrt{\left(\dfrac{1}{a}+\dfrac{1}{b}-\dfrac{1}{a+b}\right)^2}\)\(=\left|\dfrac{1}{a}+\dfrac{1}{b}-\dfrac{1}{a+b}\right|\)