Question

\(\left(1+\dfrac{\sqrt{a}}{a+1}\right):\left(\dfrac{1}{\sqrt{a-1}}-\dfrac{2\sqrt{a}}{a\sqrt{a}+\sqrt{a}-a-1}\right)\\ a,TìmađểbiểuthứcAcónghĩa.Rútgọn\\ b,TínhgiátrịcủaAkhia=\dfrac{2}{7+3\sqrt{5}}\\ c,TìmasaochoA< 1\)

Answer 1

a: ĐKXĐ: a>=0; a<>1

\(A=\dfrac{a+\sqrt{a}+1}{a+1}:\dfrac{a+1-2\sqrt{a}}{\left(\sqrt{a}-1\right)\left(a+1\right)}\)

\(=\dfrac{a+\sqrt{a}+1}{a+1}\cdot\dfrac{a+1}{\sqrt{a}-1}=\dfrac{a+\sqrt{a}+1}{\sqrt{a}-1}\)

b: Thay \(a=\dfrac{2}{7+3\sqrt{5}}=\dfrac{14-6\sqrt{5}}{4}\) vào A, ta được:

\(A=\dfrac{\dfrac{14-6\sqrt{5}}{4}+\dfrac{3-\sqrt{5}}{2}+1}{\dfrac{3-\sqrt{5}}{2}-1}\)

\(=\dfrac{14-6\sqrt{5}+6-2\sqrt{5}+4}{4}:\dfrac{1-\sqrt{5}}{2}\)

\(=\dfrac{24-8\sqrt{5}}{4}\cdot\dfrac{2}{1-\sqrt{5}}=2-2\sqrt{5}\)