Question

Cho M = \((\frac{a^2+b^2}{a^2-b^2}):[\left(\frac{a+b}{a-b}+\frac{a}{b}\right).\left(\frac{a+b}{a-b}-\frac{b}{a}\right)]\)

a, Rút gọn M.

b, Tính giá trị của M khi \(\frac{a}{b}=\sqrt{2}\).

Answer 1

ĐKXĐ:...

\(M=\left(\frac{a^2+b^2}{a^2-b^2}\right):\left[\left(\frac{a^2+b^2}{b\left(a-b\right)}\right)\left(\frac{a^2+b^2}{a\left(a-b\right)}\right)\right]=\left(\frac{a^2+b^2}{\left(a-b\right)\left(a+b\right)}\right)\left(\frac{ab\left(a-b\right)^2}{\left(a^2+b^2\right)^2}\right)=\frac{ab\left(a-b\right)}{\left(a+b\right)\left(a^2+b^2\right)}\)

b/ \(M=\frac{a-b}{\left(\frac{a}{b}+1\right)\left(a+b.\frac{b}{a}\right)}=\frac{\frac{a}{b}-1}{\left(\frac{a}{b}+1\right)\left(\frac{a}{b}+\frac{b}{a}\right)}=\frac{\sqrt{2}-1}{\left(\sqrt{2}+1\right)\left(\sqrt{2}+\frac{1}{\sqrt{2}}\right)}=\frac{3\sqrt{2}-4}{3}\)