Question

So sánh (ko dùng máy tính bỏ túi hay bảng số)

a)\(\sqrt{2}\)+\(\sqrt{3}\) và \(\sqrt{10}\)

b)\(\sqrt{3}\)+2 và \(\sqrt{2}\)+\(\sqrt{16}\)

c)16 và \(\sqrt{15}\).\(\sqrt{17}\)

d)8 và \(\sqrt{15}\)+\(\sqrt{17}\)

Answer 1

a/ \(\left(\sqrt{2}+\sqrt{3}\right)^2=2+3+2\sqrt{2.3}=5+2\sqrt{6}=5+\sqrt{24}\)

\(\left(\sqrt{10}\right)^2=10=5+5=5+\sqrt{25}\)

Vì \(\sqrt{24}< \sqrt{25}\)

=>\(\sqrt{2}+\sqrt{3}< \sqrt{10}\)

b/\(\left(\sqrt{3}+2\right)^2=3+4+4\sqrt{3}=7+4\sqrt{3}\)

\(\left(\sqrt{2}+\sqrt{16}\right)^2=2+16+2\sqrt{2.16}=18+4\sqrt{8}\)

=> \(\sqrt{3}+2< \sqrt{2}+\sqrt{16}\)

c/ \(16=\sqrt{16^2}\)

\(\sqrt{15}.\sqrt{17}=\sqrt{15.17}=\sqrt{\left(16-1\right)\left(16+1\right)}=\sqrt{16^2-1}\)

=> \(16>\sqrt{15}.\sqrt{17}\)

d/\(8^2=64=32+32=32+2\sqrt{256}\)

\(\left(\sqrt{15}+\sqrt{17}\right)^2=15+17+2\sqrt{15.17}=32+2\sqrt{255}\)

=> \(8>\sqrt{15}+\sqrt{17}\)