Question

So sánh

Answer 1

Giả sử \(\sqrt[]{2}+\sqrt[]{3}< 3\)

\(\Leftrightarrow\left(\sqrt[]{2}+\sqrt[]{3}\right)^2< 3^2\)

\(\Leftrightarrow2+3+2\sqrt[]{6}< 9\)

\(\Leftrightarrow2\sqrt[]{6}< 4\)

\(\Leftrightarrow\left(2\sqrt[]{6}\right)^2< 4^2\)

\(\Leftrightarrow24< 16\left(sai\right)\)

Vậy \(\sqrt[]{2}+\sqrt[]{3}>3\)

Answer 2

Ta có :

\(\sqrt{2}+\sqrt{3}=\left(\sqrt{2}+\sqrt{3}\right)^2\\ =\left(\sqrt{2}\right)^2+2\cdot\sqrt{2}\cdot\sqrt{3}+\left(\sqrt{3}\right)^2\\ =2+2\sqrt{6}+3\\ =5+2\sqrt{6}\left(5+2\sqrt{6}>5+2\sqrt{4}\right)\)

Mà \(5+2\sqrt{4}=5+2\cdot2=5+4=9\)

\(\Rightarrow5+2\sqrt{6}>9\)

Nên \(\sqrt{2}+\sqrt{3}>3\)