Question

[Vân Bùi Khánh]

so sánh:
\(\sqrt{7}-\sqrt{5}và\sqrt{5}-\sqrt{3}\)
\(\frac{1}{3}\sqrt{6}và6\sqrt{\frac{1}{3}}\)

Answer 1

[Nguyễn Việt Lâm]

\(\sqrt{7}-\sqrt{5}=\frac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{\sqrt{7}+\sqrt{5}}=\frac{2}{\sqrt{7}+\sqrt{5}}\)

\(\sqrt{5}-\sqrt{3}=\frac{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}{\sqrt{5}+\sqrt{3}}=\frac{2}{\sqrt{5}+\sqrt{3}}\)

Do \(\sqrt{7}>\sqrt{3}\Rightarrow\sqrt{7}+\sqrt{5}>\sqrt{5}+\sqrt{3}\Rightarrow\frac{2}{\sqrt{7}+\sqrt{5}}< \frac{2}{\sqrt{5}+\sqrt{3}}\)

\(\Rightarrow\sqrt{7}-\sqrt{5}< \sqrt{5}-\sqrt{3}\)

b/ \(\frac{1}{3}\sqrt{6}=\sqrt{\frac{2}{3}}\) ; \(6\sqrt{\frac{1}{3}}=2\sqrt{3}=\sqrt{12}\)

Mà \(12>\frac{2}{3}\Rightarrow\sqrt{12}>\sqrt{\frac{2}{3}}\Rightarrow\frac{1}{3}\sqrt{6}< 6\sqrt{\frac{1}{3}}\)