Question

So sánh:

a/√2+√3 và 2

b/√24+√45 và 12

Answer 1

a/ Ta có: \(\left(\sqrt{2}+\sqrt{3}\right)^2=2+2\sqrt{6}+3=5+2\sqrt{6}\)

Vì \(2^2=4< 5\Rightarrow2^2< 5+2\sqrt{6}\Rightarrow2^2< \left(\sqrt{2}+\sqrt{3}\right)^2\)

Do đó : \(\sqrt{2}+\sqrt{3}>2\)

b/ Ta có: \((\sqrt{24}+\sqrt{45})^2=24+45+2\sqrt{4.6.9.5}=69+12\sqrt{30}\)

\(12^2=144=69+75\)

Lại có: \(\left(12\sqrt{30}\right)^2=144.30=4320\)

\(75^2=5625\)

Vì \(4320< 5625\Rightarrow12\sqrt{30}< 75\Rightarrow12\sqrt{30}+69< 75+69\Rightarrow\left(\sqrt{24}+\sqrt{45}\right)^2< 12^2\Rightarrow\sqrt{24}+\sqrt{45}< 12\)