\(P=\left(\frac{1}{\sqrt{a}}+\frac{\sqrt{a}}{\sqrt{a}+1}\right):\left(\frac{\sqrt{a}}{a+\sqrt{a}}\right)\)
\(=\frac{\sqrt{a}+1+a}{\sqrt{a}\left(\sqrt{a}+1\right)}:\left(\frac{\sqrt{a}}{\sqrt{a}\left(\sqrt{a}+1\right)}\right)=\frac{\left(a+\sqrt{a}+1\right)\left(\sqrt{a}+1\right)}{\sqrt{a}\left(\sqrt{a}+1\right)}=\frac{a+\sqrt{a}+1}{\sqrt{a}}\)
\(a=\frac{2}{\sqrt{5}-1}-\frac{2}{\sqrt{5}+1}=\frac{2\sqrt{5}+2-2\sqrt{5}+2}{\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)}=\frac{4}{4}=1\)
\(\Rightarrow P=3\)
\(P=\frac{a+\sqrt{a}+1}{\sqrt{a}}=\sqrt{a}+\frac{1}{\sqrt{a}}+1\ge2\sqrt{\frac{\sqrt{a}}{\sqrt{a}}}+1=3\)
\(\Rightarrow P_{min}=3\) khi \(a=1\)
