Question

so sánh a) \(2\sqrt{2}\)-1 và 2

b)\(2\sqrt{5}\)- \(5\sqrt{2}\) và 1

Answer 1

a) Ta có: \(\left(2\sqrt{2}-1\right)^2=8-2\cdot2\sqrt{2}\cdot1+1=9-4\sqrt{2}=4+5-4\sqrt{2}\)

\(2^2=4=4+0\)

Ta có: \(5=\sqrt{25}\)

\(4\sqrt{2}=\sqrt{32}\)

mà \(\sqrt{25}< \sqrt{32}\)

nên \(5< 4\sqrt{2}\)

\(\Leftrightarrow5-4\sqrt{2}< 0\)

\(\Leftrightarrow5-4\sqrt{2}+4< 0+4\)

\(\Leftrightarrow9-4\sqrt{2}< 4\)

\(\Leftrightarrow\left(2\sqrt{2}-1\right)^2< 2^2\)

hay \(2\sqrt{2}-1< 2\)

b) Ta có: \(2\sqrt{5}=\sqrt{4\cdot5}=\sqrt{20}\)

\(5\sqrt{2}=\sqrt{25\cdot2}=\sqrt{50}\)

mà \(\sqrt{20}< \sqrt{50}\)

nên \(2\sqrt{5}< 5\sqrt{2}\)

\(\Leftrightarrow2\sqrt{5}-5\sqrt{2}< 0\)

mà 0<1

nên \(2\sqrt{5}-5\sqrt{2}< 1\)