Question

So sánh:

A = \(\sqrt{4-2\sqrt{3}}+3\sqrt{3}\)

B = \(\frac{5\sqrt{5}+2\sqrt{2}}{\sqrt{5}+\sqrt{2}}+\frac{\sqrt{10}+10}{\sqrt{10}+1}\)

Answer 1

\(A=\sqrt{3-2\sqrt{3}+1}+3\sqrt{3}=\sqrt{\sqrt{3}^2-2.\sqrt{3}+1}+3\sqrt{3}=\sqrt{\left(\sqrt{3}-1\right)^2}+3\sqrt{3}=\sqrt{3}-1+3\sqrt{3}\left(do:\sqrt{3}>\sqrt{1}=1\right)=4\sqrt{3}-1\) \(B=\frac{5\sqrt{5}+2\sqrt{2}}{\sqrt{5}+\sqrt{2}}+\frac{\sqrt{10}+10}{\sqrt{10}+1}=\frac{\sqrt{5^3}+\sqrt{2^3}}{\sqrt{5}+\sqrt{2}}+\frac{\sqrt{10}\left(\sqrt{10}+1\right)}{\sqrt{10}+1}=\frac{\left(\sqrt{5}\right)^3+\left(\sqrt{2}\right)^3}{\sqrt{5}+\sqrt{2}}+\sqrt{10}=\frac{\left(\sqrt{5}+\sqrt{2}\right)\left(7-\sqrt{10}\right)}{\sqrt{5}+\sqrt{2}}+\sqrt{10}=7=8-1=4.2-1=4.\sqrt{4}-1>4\sqrt{3}-1=A\)

\(\Rightarrow B>A\)