Question

[阮芳草]

Đưa thừa số vào trong dấu căn rồi so sánh các cặp số

a)\(2\sqrt{5}\) và \(5\sqrt{2}\)

b) \(3\sqrt{13}\)và \(4\sqrt{11}\)

c) \(\frac{3}{4}.\sqrt{7}\)và \(\frac{2}{5}.\sqrt{5}\)

d) \(\frac{2}{a-b}.\sqrt{\frac{a^2-b^2}{2}}\) ( với 0 < a < b ) 

Answer 1

[阮芳草]

a)Ta có:  \(2\sqrt{5}< 5\sqrt{2}\)\(2\sqrt{5}=\sqrt{2^2.5}=\sqrt{20}\)

\(5\sqrt{2}=\sqrt{5^2.2}=\sqrt{50}\)

Vì \(\sqrt{20}< \sqrt{50}\)

Nên \(2\sqrt{5}< 5\sqrt{2}\)

b)Ta có: \(3\sqrt{13}=\sqrt{3^2.13}=\sqrt{117}\)

\(4\sqrt{11}=\sqrt{4^2.11}=\sqrt{176}\)

Vì \(\sqrt{117}< \sqrt{176}\)

Nên \(3\sqrt{13}< 4\sqrt{11}\)

c) Ta có: \(\frac{3}{4}.\sqrt{7}=\sqrt{\left(\frac{3}{4}\right)^2.7}=\sqrt{\frac{63}{16}}\)

\(\frac{2}{5}.\sqrt{5}=\sqrt{\left(\frac{2}{5}\right)^2.5}=\sqrt{\frac{4}{5}}\)

Vì \(\sqrt{\frac{63}{16}}>1\)

\(\sqrt{\frac{4}{5}}< 1\)

Nên \(\sqrt{\frac{63}{16}}>\sqrt{\frac{4}{5}}\)

Vậy \(\frac{3}{4}.\sqrt{7}>\frac{2}{5}.\sqrt{5}\)